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The Matrix Cookbook
皓 谷The Matrix Cookbook
[ http://matrixcookbook.com ]
Kaare Brandt Petersen
Michael Syskind Pedersen
Version: November 15, 2012
1
Introduction
What is this? These pages are a collection of facts (identities, approximations, inequalities, relations, ...) about matrices and matters relating to them.
It is collected in this form for the convenience of anyone who wants a quick
desktop reference .
Disclaimer: The identities, approximations and relations presented here were
obviously not invented but collected, borrowed and copied from a large amount
of sources. These sources include similar but shorter notes found on the internet
and appendices in books - see the references for a full list.
Errors: Very likely there are errors, typos, and mistakes for which we apologize and would be grateful to receive corrections at cookbook@2302.dk.
Its ongoing: The project of keeping a large repository of relations involving
matrices is naturally ongoing and the version will be apparent from the date in
the header.
Suggestions: Your suggestion for additional content or elaboration of some
topics is most welcome acookbook@2302.dk.
Keywords: Matrix algebra, matrix relations, matrix identities, derivative of
determinant, derivative of inverse matrix, differentiate a matrix.
Acknowledgements: We would like to thank the following for contributions
and suggestions: Bill Baxter, Brian Templeton, Christian Rishøj, Christian
Schröppel, Dan Boley, Douglas L. Theobald, Esben Hoegh-Rasmussen, Evripidis
Karseras, Georg Martius, Glynne Casteel, Jan Larsen, Jun Bin Gao, Jürgen
Struckmeier, Kamil Dedecius, Karim T. Abou-Moustafa, Korbinian Strimmer,
Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut,
Markus Froeb, Michael Hubatka, Miguel Barão, Ole Winther, Pavel Sakov,
Stephan Hattinger, Troels Pedersen, Vasile Sima, Vincent Rabaud, Zhaoshui
He. We would also like thank The Oticon Foundation for funding our PhD
studies.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 2
CONTENTS
CONTENTS
Contents
1 Basics
1.1 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The Special Case 2x2 . . . . . . . . . . . . . . . . . . . . . . . . .
2 Derivatives
2.1 Derivatives
2.2 Derivatives
2.3 Derivatives
2.4 Derivatives
2.5 Derivatives
2.6 Derivatives
2.7 Derivatives
2.8 Derivatives
of
of
of
of
of
of
of
of
a Determinant . . . . . . . . . . . .
an Inverse . . . . . . . . . . . . . . .
Eigenvalues . . . . . . . . . . . . . .
Matrices, Vectors and Scalar Forms
Traces . . . . . . . . . . . . . . . . .
vector norms . . . . . . . . . . . . .
matrix norms . . . . . . . . . . . . .
Structured Matrices . . . . . . . . .
3 Inverses
3.1 Basic . . . . . . . . . . .
3.2 Exact Relations . . . . .
3.3 Implication on Inverses .
3.4 Approximations . . . . .
3.5 Generalized Inverse . . .
3.6 Pseudo Inverse . . . . .
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7
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8
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12
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14
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17
17
18
20
20
21
21
4 Complex Matrices
24
4.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Higher order and non-linear derivatives . . . . . . . . . . . . . . . 26
4.3 Inverse of complex sum . . . . . . . . . . . . . . . . . . . . . . . 27
5 Solutions and Decompositions
5.1 Solutions to linear equations .
5.2 Eigenvalues and Eigenvectors
5.3 Singular Value Decomposition
5.4 Triangular Decomposition . .
5.5 LU decomposition . . . . . .
5.6 LDM decomposition . . . . .
5.7 LDL decompositions . . . . .
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28
28
30
31
32
32
33
33
6 Statistics and Probability
34
6.1 Definition of Moments . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2 Expectation of Linear Combinations . . . . . . . . . . . . . . . . 35
6.3 Weighted Scalar Variable . . . . . . . . . . . . . . . . . . . . . . 36
7 Multivariate Distributions
7.1 Cauchy . . . . . . . . . .
7.2 Dirichlet . . . . . . . . . .
7.3 Normal . . . . . . . . . .
7.4 Normal-Inverse Gamma .
7.5 Gaussian . . . . . . . . . .
7.6 Multinomial . . . . . . . .
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37
37
37
37
37
37
37
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 3
CONTENTS
7.7
7.8
7.9
CONTENTS
Student’s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wishart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wishart, Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Gaussians
8.1 Basics . . . . . . . .
8.2 Moments . . . . . .
8.3 Miscellaneous . . . .
8.4 Mixture of Gaussians
37
38
39
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40
40
42
44
44
9 Special Matrices
9.1 Block matrices . . . . . . . . . . . . . . . .
9.2 Discrete Fourier Transform Matrix, The . .
9.3 Hermitian Matrices and skew-Hermitian . .
9.4 Idempotent Matrices . . . . . . . . . . . . .
9.5 Orthogonal matrices . . . . . . . . . . . . .
9.6 Positive Definite and Semi-definite Matrices
9.7 Singleentry Matrix, The . . . . . . . . . . .
9.8 Symmetric, Skew-symmetric/Antisymmetric
9.9 Toeplitz Matrices . . . . . . . . . . . . . . .
9.10 Transition matrices . . . . . . . . . . . . . .
9.11 Units, Permutation and Shift . . . . . . . .
9.12 Vandermonde Matrices . . . . . . . . . . . .
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46
46
47
48
49
49
50
52
54
54
55
56
57
10 Functions and Operators
10.1 Functions and Series . . . . .
10.2 Kronecker and Vec Operator
10.3 Vector Norms . . . . . . . . .
10.4 Matrix Norms . . . . . . . . .
10.5 Rank . . . . . . . . . . . . . .
10.6 Integral Involving Dirac Delta
10.7 Miscellaneous . . . . . . . . .
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58
58
59
61
61
62
62
63
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Functions
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A One-dimensional Results
64
A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 65
B Proofs and Details
66
B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 4
CONTENTS
CONTENTS
Notation and Nomenclature
A
Aij
Ai
Aij
An
A−1
A+
A1/2
(A)ij
Aij
[A]ij
a
ai
ai
a
ℜz
ℜz
ℜZ
ℑz
ℑz
ℑZ
Matrix
Matrix indexed for some purpose
Matrix indexed for some purpose
Matrix indexed for some purpose
Matrix indexed for some purpose or
The n.th power of a square matrix
The inverse matrix of the matrix A
The pseudo inverse matrix of the matrix A (see Sec. 3.6)
The square root of a matrix (if unique), not elementwise
The (i, j).th entry of the matrix A
The (i, j).th entry of the matrix A
The ij-submatrix, i.e. A with i.th row and j.th column deleted
Vector (column-vector)
Vector indexed for some purpose
The i.th element of the vector a
Scalar
Real part of a scalar
Real part of a vector
Real part of a matrix
Imaginary part of a scalar
Imaginary part of a vector
Imaginary part of a matrix
det(A)
Tr(A)
diag(A)
eig(A)
vec(A)
sup
||A||
AT
A−T
A∗
AH
Determinant of A
Trace of the matrix A
Diagonal matrix of the matrix A, i.e. (diag(A))ij = δij Aij
Eigenvalues of the matrix A
The vector-version of the matrix A (see Sec. 10.2.2)
Supremum of a set
Matrix norm (subscript if any denotes what norm)
Transposed matrix
The inverse of the transposed and vice versa, A−T = (A−1 )T = (AT )−1 .
Complex conjugated matrix
Transposed and complex conjugated matrix (Hermitian)
A◦B
A⊗B
Hadamard (elementwise) product
Kronecker product
0
I
Jij
Σ
Λ
The null matrix. Zero in all entries.
The identity matrix
The single-entry matrix, 1 at (i, j) and zero elsewhere
A positive definite matrix
A diagonal matrix
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 5
1 BASICS
1
1.1
1.2
Basics
(AB)−1
(ABC...)−1
=
=
B−1 A−1
...C−1 B−1 A−1
(1)
(2)
(AT )−1
(A + B)T
=
=
(A−1 )T
A T + BT
(3)
(4)
(AB)T
=
BT A T
T
(5)
T
T
T
(ABC...)
(AH )−1
(A + B)H
=
=
=
...C B A
(A−1 )H
AH + BH
(AB)H
(ABC...)H
=
=
BH A H
...CH BH AH
Tr(A)
=
Tr(A)
Tr(A)
=
=
P
Tr(A )
(12)
(13)
Tr(AB)
=
Tr(BA)
(14)
Tr(A + B)
Tr(ABC)
=
=
Tr(A) + Tr(B)
Tr(BCA) = Tr(CAB)
(15)
(16)
aT a
=
Tr(aaT )
(17)
(6)
(7)
(8)
(9)
(10)
Trace
Pi
Aii
(11)
i λi ,
T
λi = eig(A)
Determinant
Let A be an n × n matrix.
det(A)
=
det(cA)
det(AT )
=
=
Q
i λi
λi = eig(A)
cn det(A),
det(A)
if A ∈ Rn×n
(18)
(19)
(20)
det(AB)
=
det(A) det(B)
(21)
det(A−1 )
det(An )
=
=
1/ det(A)
det(A)n
(22)
(23)
det(I + uvT )
=
1 + uT v
(24)
det(I + A) = 1 + det(A) + Tr(A)
(25)
1
1
det(I + A) = 1 + det(A) + Tr(A) + Tr(A)2 − Tr(A2 )
2
2
(26)
For n = 2:
For n = 3:
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 6
1.3
The Special Case 2x2
1 BASICS
For n = 4:
det(I + A)
=
1 + det(A) + Tr(A) +
1
2
1
+Tr(A)2 − Tr(A2 )
2
1
1
1
3
+ Tr(A) − Tr(A)Tr(A2 ) + Tr(A3 )
6
2
3
(27)
For small ε, the following approximation holds
1
1
det(I + εA) ∼
= 1 + det(A) + εTr(A) + ε2 Tr(A)2 − ε2 Tr(A2 )
2
2
1.3
(28)
The Special Case 2x2
Consider the matrix A
A=
A11
A21
A12
A22
Determinant and trace
det(A) = A11 A22 − A12 A21
(29)
Tr(A) = A11 + A22
(30)
Eigenvalues
λ2 − λ · Tr(A) + det(A) = 0
λ1 =
Tr(A) +
p
Tr(A)2 − 4 det(A)
2
λ1 + λ2 = Tr(A)
λ2 =
Tr(A) −
p
Tr(A)2 − 4 det(A)
2
λ1 λ2 = det(A)
Eigenvectors
v1 ∝
A12
λ1 − A11
Inverse
A
−1
1
=
det(A)
v2 ∝
A22
−A21
A12
λ2 − A11
−A12
A11
(31)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 7
2 DERIVATIVES
2
Derivatives
This section is covering differentiation of a number of expressions with respect to
a matrix X. Note that it is always assumed that X has no special structure, i.e.
that the elements of X are independent (e.g. not symmetric, Toeplitz, positive
definite). See section 2.8 for differentiation of structured matrices. The basic
assumptions can be written in a formula as
∂Xkl
= δik δlj
∂Xij
that is for e.g. vector forms,
∂xi
∂x
=
∂y i
∂y
∂x
∂y
=
i
(32)
∂x
∂yi
∂x
∂y
=
ij
∂xi
∂yj
The following rules are general and very useful when deriving the differential of
an expression ([19]):
∂A
∂(αX)
∂(X + Y)
∂(Tr(X))
∂(XY)
∂(X ◦ Y)
∂(X ⊗ Y)
∂(X−1 )
∂(det(X))
∂(det(X))
∂(ln(det(X)))
∂XT
∂XH
2.1
2.1.1
=
=
=
=
=
=
=
=
=
=
=
=
=
0
α∂X
∂X + ∂Y
Tr(∂X)
(∂X)Y + X(∂Y)
(∂X) ◦ Y + X ◦ (∂Y)
(∂X) ⊗ Y + X ⊗ (∂Y)
−X−1 (∂X)X−1
Tr(adj(X)∂X)
det(X)Tr(X−1 ∂X)
Tr(X−1 ∂X)
(∂X)T
(∂X)H
(A is a constant)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
Derivatives of a Determinant
General form
∂ det(Y)
∂x
X ∂ det(X)
Xjk
∂Xik
=
det(Y)Tr Y
=
δij det(X)
−1 ∂Y
∂x
(46)
(47)
k
∂ 2 det(Y)
∂x2
=
"
"
det(Y) Tr Y
∂Y
−1 ∂ ∂x
#
∂x
∂Y
∂Y
Tr Y−1
+Tr Y−1
∂x
∂x
#
−1 ∂Y
−1 ∂Y
Y
−Tr Y
∂x
∂x
(48)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 8
2.2
Derivatives of an Inverse
2.1.2
2 DERIVATIVES
Linear forms
∂ det(X)
∂X
X ∂ det(X)
Xjk
∂Xik
=
det(X)(X−1 )T
(49)
=
δij det(X)
(50)
=
det(AXB)(X−1 )T = det(AXB)(XT )−1
(51)
k
∂ det(AXB)
∂X
2.1.3
Square forms
If X is square and invertible, then
∂ det(XT AX)
= 2 det(XT AX)X−T
∂X
(52)
If X is not square but A is symmetric, then
∂ det(XT AX)
= 2 det(XT AX)AX(XT AX)−1
∂X
(53)
If X is not square and A is not symmetric, then
∂ det(XT AX)
= det(XT AX)(AX(XT AX)−1 + AT X(XT AT X)−1 )
∂X
2.1.4
(54)
Other nonlinear forms
Some special cases are (See [9, 7])
∂ ln det(XT X)|
∂X
∂ ln det(XT X)
∂X+
∂ ln | det(X)|
∂X
∂ det(Xk )
∂X
2.2
=
2(X+ )T
(55)
=
−2XT
(56)
=
(X−1 )T = (XT )−1
(57)
=
k det(Xk )X−T
(58)
Derivatives of an Inverse
From [27] we have the basic identity
∂Y −1
∂Y−1
= −Y−1
Y
∂x
∂x
(59)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 9
2.3
Derivatives of Eigenvalues
2 DERIVATIVES
from which it follows
∂(X−1 )kl
∂Xij
=
−(X−1 )ki (X−1 )jl
(60)
∂aT X−1 b
= −X−T abT X−T
(61)
∂X
∂ det(X−1 )
= − det(X−1 )(X−1 )T
(62)
∂X
∂Tr(AX−1 B)
= −(X−1 BAX−1 )T
(63)
∂X
∂Tr((X + A)−1 )
= −((X + A)−1 (X + A)−1 )T
(64)
∂X
From [32] we have the following result: Let A be an n × n invertible square
matrix, W be the inverse of A, and J(A) is an n × n -variate and differentiable
function with respect to A, then the partial differentials of J with respect to A
and W satisfy
∂J −T
∂J
= −A−T
A
∂A
∂W
2.3
Derivatives of Eigenvalues
∂
∂ X
Tr(X) = I
(65)
eig(X) =
∂X
∂X
∂ Y
∂
det(X) = det(X)X−T
(66)
eig(X) =
∂X
∂X
If A is real and symmetric, λi and vi are distinct eigenvalues and eigenvectors
of A (see (276)) with viT vi = 1, then [33]
∂λi
∂vi
2.4
2.4.1
=
=
viT ∂(A)vi
(67)
+
(λi I − A) ∂(A)vi
(68)
Derivatives of Matrices, Vectors and Scalar Forms
First Order
∂xT a
∂x
∂aT Xb
∂X
∂aT XT b
∂X
∂aT Xa
∂X
∂X
∂Xij
∂(XA)ij
∂Xmn
∂(XT A)ij
∂Xmn
=
∂aT x
∂x
=
abT
(70)
=
baT
(71)
=
∂aT XT a
∂X
=
Jij
=
δim (A)nj
=
(Jmn A)ij
(74)
=
δin (A)mj
=
(Jnm A)ij
(75)
=
a
=
(69)
aaT
(72)
(73)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 10
2.4
Derivatives of Matrices, Vectors and Scalar Forms
2.4.2
2 DERIVATIVES
Second Order
∂ X
Xkl Xmn
∂Xij
(76)
=
X(bcT + cbT )
(77)
=
BT C(Dx + d) + DT CT (Bx + b)
(78)
=
δlj (XT B)ki + δkj (BX)il
(79)
∂(XT BX)
∂Xij
=
XT BJij + Jji BX
2
kl
klmn
∂bT XT Xc
∂X
∂(Bx + b)T C(Dx + d)
∂x
∂(XT BX)kl
∂Xij
X
Xkl
=
(Jij )kl = δik δjl (80)
See Sec 9.7 for useful properties of the Single-entry matrix Jij
∂xT Bx
∂x
∂bT XT DXc
∂X
∂
(Xb + c)T D(Xb + c)
∂X
=
(B + BT )x
(81)
=
DT XbcT + DXcbT
(82)
=
(D + DT )(Xb + c)bT
(83)
=
−2AT W(x − As)
(84)
=
2W(x − s)
(85)
=
−2W(x − s)
(86)
=
2W(x − As)
(87)
=
−2W(x − As)sT
(88)
Assume W is symmetric, then
∂
(x − As)T W(x − As)
∂s
∂
(x − s)T W(x − s)
∂x
∂
(x − s)T W(x − s)
∂s
∂
(x − As)T W(x − As)
∂x
∂
(x − As)T W(x − As)
∂A
As a case with complex values the following holds
∂(a − xH b)2
∂x
=
−2b(a − xH b)∗
(89)
This formula is also known from the LMS algorithm [14]
2.4.3
Higher-order and non-linear
n−1
X
∂(Xn )kl
(Xr Jij Xn−1−r )kl
=
∂Xij
r=0
(90)
For proof of the above, see B.1.3.
n−1
X
∂ T n
(Xr )T abT (Xn−1−r )T
a X b=
∂X
r=0
(91)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 11
2.5
Derivatives of Traces
∂ T n T n
a (X ) X b
∂X
2 DERIVATIVES
n−1
Xh
=
Xn−1−r abT (Xn )T Xr
r=0
+(Xr )T Xn abT (Xn−1−r )T
i
(92)
See B.1.3 for a proof.
Assume s and r are functions of x, i.e. s = s(x), r = r(x), and that A is a
constant, then
T
T
∂r
∂s
∂ T
Ar +
AT s
(93)
s Ar =
∂x
∂x
∂x
∂ (Ax)T (Ax)
∂x (Bx)T (Bx)
=
=
2.4.4
∂ xT AT Ax
∂x xT BT Bx
xT AT AxBT Bx
AT Ax
−2
2 T
x BBx
(xT BT Bx)2
(94)
(95)
Gradient and Hessian
Using the above we have for the gradient and the Hessian
f
∂f
∇x f =
∂x
∂2f
∂x∂xT
2.5
=
xT Ax + bT x
(96)
=
(A + AT )x + b
(97)
=
A + AT
(98)
Derivatives of Traces
Assume F (X) to be a differentiable function of each of the elements of X. It
then holds that
∂Tr(F (X))
= f (X)T
∂X
where f (·) is the scalar derivative of F (·).
2.5.1
First Order
∂
Tr(X)
∂X
∂
Tr(XA)
∂X
∂
Tr(AXB)
∂X
∂
Tr(AXT B)
∂X
∂
Tr(XT A)
∂X
∂
Tr(AXT )
∂X
∂
Tr(A ⊗ X)
∂X
=
I
(99)
=
AT
(100)
=
A T BT
(101)
=
BA
(102)
=
A
(103)
=
A
(104)
=
Tr(A)I
(105)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 12
2.5
Derivatives of Traces
2.5.2
2 DERIVATIVES
Second Order
∂
Tr(X2 )
∂X
∂
Tr(X2 B)
∂X
∂
Tr(XT BX)
∂X
∂
Tr(BXXT )
∂X
∂
Tr(XXT B)
∂X
∂
Tr(XBXT )
∂X
∂
Tr(BXT X)
∂X
∂
Tr(XT XB)
∂X
∂
Tr(AXBX)
∂X
∂
Tr(XT X)
∂X
∂
Tr(BT XT CXB)
∂X
∂
Tr XT BXC
∂X
∂
Tr(AXBXT C)
∂X
h
i
∂
Tr (AXB + C)(AXB + C)T
∂X
∂
Tr(X ⊗ X)
∂X
=
2XT
(106)
=
(XB + BX)T
(107)
=
BX + BT X
(108)
=
BX + BT X
(109)
=
BX + BT X
(110)
=
XBT + XB
(111)
=
XBT + XB
(112)
=
XBT + XB
(113)
=
AT XT BT + BT XT AT
(114)
=
∂
Tr(XXT )
∂X
(115)
=
CT XBBT + CXBBT
(116)
=
BXC + BT XCT
(117)
=
AT CT XBT + CAXB
(118)
=
2AT (AXB + C)BT
(119)
=
∂
Tr(X)Tr(X) = 2Tr(X)I(120)
∂X
=
k(Xk−1 )T
=
2X
See [7].
2.5.3
Higher Order
∂
Tr(Xk )
∂X
∂
Tr(AXk )
∂X
T T
T
∂
∂X Tr B X CXX CXB
=
k−1
X
(Xr AXk−r−1 )T
(121)
(122)
r=0
=
CXXT CXBBT
+CT XBBT XT CT X
+CXBBT XT CX
+CT XXT CT XBBT
(123)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 13
2.6
Derivatives of vector norms
2 DERIVATIVES
2.5.4
Other
∂
Tr(AX−1 B) = −(X−1 BAX−1 )T = −X−T AT BT X−T
∂X
Assume B and C to be symmetric, then
h
i
∂
Tr (XT CX)−1 A
∂X
h
i
∂
Tr (XT CX)−1 (XT BX)
∂X
∂
Tr (A + XT CX)−1 (XT BX)
∂X
h
i
(124)
=
−(CX(XT CX)−1 )(A + AT )(XT CX)−1 (125)
=
−2CX(XT CX)−1 XT BX(XT CX)−1
+2BX(XT CX)−1
=
(126)
−2CX(A + XT CX)−1 XT BX(A + XT CX)−1
+2BX(A + XT CX)−1
(127)
See [7].
∂Tr(sin(X))
∂X
2.6
2.6.1
=
cos(X)T
Derivatives of vector norms
Two-norm
∂
x−a
||x − a||2 =
∂x
||x − a||2
I
(x − a)(x − a)T
∂ x−a
=
−
∂x kx − ak2
kx − ak2
kx − ak32
∂||xT x||2
∂||x||22
=
= 2x
∂x
∂x
2.7
(128)
(129)
(130)
(131)
Derivatives of matrix norms
For more on matrix norms, see Sec. 10.4.
2.7.1
Frobenius norm
∂
∂
||X||2F =
Tr(XXH ) = 2X
(132)
∂X
∂X
See (248). Note that this is also a special case of the result in equation 119.
2.8
Derivatives of Structured Matrices
Assume that the matrix A has some structure, i.e. symmetric, toeplitz, etc.
In that case the derivatives of the previous section does not apply in general.
Instead, consider the following general rule for differentiating a scalar function
f (A)
#
"
T
X ∂f ∂Akl
∂A
∂f
df
(133)
=
= Tr
dAij
∂Akl ∂Aij
∂A
∂Aij
kl
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 14
2.8
Derivatives of Structured Matrices
2 DERIVATIVES
The matrix differentiated with respect to itself is in this document referred to
as the structure matrix of A and is defined simply by
∂A
= Sij
∂Aij
(134)
If A has no special structure we have simply Sij = Jij , that is, the structure
matrix is simply the single-entry matrix. Many structures have a representation
in singleentry matrices, see Sec. 9.7.6 for more examples of structure matrices.
2.8.1
The Chain Rule
Sometimes the objective is to find the derivative of a matrix which is a function
of another matrix. Let U = f (X), the goal is to find the derivative of the
function g(U) with respect to X:
∂g(f (X))
∂g(U)
=
∂X
∂X
(135)
Then the Chain Rule can then be written the following way:
M
N
∂g(U)
∂g(U) X X ∂g(U) ∂ukl
=
=
∂X
∂xij
∂ukl ∂xij
(136)
k=1 l=1
Using matrix notation, this can be written as:
h ∂g(U)
∂g(U)
∂U i
= Tr (
)T
.
∂Xij
∂U
∂Xij
2.8.2
(137)
Symmetric
If A is symmetric, then Sij = Jij + Jji − Jij Jij and therefore
T
∂f
df
∂f
∂f
+
=
− diag
dA
∂A
∂A
∂A
(138)
That is, e.g., ([5]):
∂Tr(AX)
∂X
∂ det(X)
∂X
∂ ln det(X)
∂X
2.8.3
=
A + AT − (A ◦ I), see (142)
(139)
=
det(X)(2X−1 − (X−1 ◦ I))
(140)
=
2X−1 − (X−1 ◦ I)
(141)
Diagonal
If X is diagonal, then ([19]):
∂Tr(AX)
∂X
=
A◦I
(142)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 15
2.8
Derivatives of Structured Matrices
2.8.4
2 DERIVATIVES
Toeplitz
Like symmetric matrices and diagonal matrices also Toeplitz matrices has a
special structure which should be taken into account when the derivative with
respect to a matrix with Toeplitz structure.
=
=
≡
∂Tr(AT)
∂T
∂Tr(TA)
∂T
(143)
Tr(A)
Tr([AT ]n1 )
Tr([AT ]1n ))
Tr(A)
.
.
Tr([[AT ]1n ]2,n−1 )
.
.
.
A1n
Tr([[AT ]1n ]n−1,2 )
.
.
.
.
.
.
.
···
.
.
.
.
.
.
.
.
.
Tr([[AT ]1n ]2,n−1 )
.
.
.
An1
···
.
.
.
.
.
Tr([[AT ]1n ]n−1,2 )
.
Tr([AT ]n1 )
Tr(A)
Tr([AT ]1n ))
α(A)
As it can be seen, the derivative α(A) also has a Toeplitz structure. Each value
in the diagonal is the sum of all the diagonal valued in A, the values in the
diagonals next to the main diagonal equal the sum of the diagonal next to the
main diagonal in AT . This result is only valid for the unconstrained Toeplitz
matrix. If the Toeplitz matrix also is symmetric, the same derivative yields
∂Tr(AT)
∂Tr(TA)
=
= α(A) + α(A)T − α(A) ◦ I
∂T
∂T
(144)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 16
3 INVERSES
3
3.1
3.1.1
Inverses
Basic
Definition
The inverse A−1 of a matrix A ∈ Cn×n is defined such that
AA−1 = A−1 A = I,
(145)
where I is the n × n identity matrix. If A−1 exists, A is said to be nonsingular.
Otherwise, A is said to be singular (see e.g. [12]).
3.1.2
Cofactors and Adjoint
The submatrix of a matrix A, denoted by [A]ij is a (n − 1) × (n − 1) matrix
obtained by deleting the ith row and the jth column of A. The (i, j) cofactor
of a matrix is defined as
cof(A, i, j) = (−1)i+j det([A]ij ),
The matrix of cofactors can be created from the cofactors
cof(A, 1, 1)
···
cof(A, 1, n)
..
..
cof(A) =
.
cof(A,
i,
j)
.
cof(A, n, 1)
···
cof(A, n, n)
(146)
(147)
The adjoint matrix is the transpose of the cofactor matrix
adj(A) = (cof(A))T ,
3.1.3
(148)
Determinant
The determinant of a matrix A ∈ Cn×n is defined as (see [12])
det(A)
=
=
n
X
j=1
n
X
(−1)j+1 A1j det ([A]1j )
(149)
A1j cof(A, 1, j).
(150)
j=1
3.1.4
Construction
The inverse matrix can be constructed, using the adjoint matrix, by
A−1 =
1
· adj(A)
det(A)
(151)
For the case of 2 × 2 matrices, see section 1.3.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 17
3.2
Exact Relations
3.1.5
3 INVERSES
Condition number
The condition number of a matrix c(A) is the ratio between the largest and the
smallest singular value of a matrix (see Section 5.3 on singular values),
c(A) =
d+
d−
(152)
The condition number can be used to measure how singular a matrix is. If the
condition number is large, it indicates that the matrix is nearly singular. The
condition number can also be estimated from the matrix norms. Here
c(A) = kAk · kA−1 k,
(153)
where k · k is a norm such as e.g the 1-norm, the 2-norm, the ∞-norm or the
Frobenius norm (see Sec 10.4p
for more on matrix norms).
The 2-norm of A equals (max(eig(AH A))) [12, p.57]. For a symmetric
matrix, this reduces to ||A||2 = max(|eig(A)|) [12, p.394]. If the matrix is
symmetric and positive definite, ||A||2 = max(eig(A)). The condition number
based on the 2-norm thus reduces to
kAk2 kA−1 k2 = max(eig(A)) max(eig(A−1 )) =
3.2
3.2.1
max(eig(A))
.
min(eig(A))
Exact Relations
Basic
(AB)−1 = B−1 A−1
3.2.2
(154)
(155)
The Woodbury identity
The Woodbury identity comes in many variants. The latter of the two can be
found in [12]
(A + CBCT )−1
(A + UBV)−1
=
=
A−1 − A−1 C(B−1 + CT A−1 C)−1 CT A−1 (156)
A−1 − A−1 U(B−1 + VA−1 U)−1 VA−1
(157)
If P, R are positive definite, then (see [30])
(P−1 + BT R−1 B)−1 BT R−1 = PBT (BPBT + R)−1
3.2.3
(158)
The Kailath Variant
(A + BC)−1 = A−1 − A−1 B(I + CA−1 B)−1 CA−1
(159)
See [4, page 153].
3.2.4
Sherman-Morrison
(A + bcT )−1 = A−1 −
A−1 bcT A−1
1 + cT A−1 b
(160)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 18
3.2
Exact Relations
3.2.5
3 INVERSES
The Searle Set of Identities
The following set of identities, can be found in [25, page 151],
3.2.6
(I + A−1 )−1
(A + BBT )−1 B
=
=
A(A + I)−1
A−1 B(I + BT A−1 B)−1
(161)
(162)
(A−1 + B−1 )−1
A − A(A + B)−1 A
=
=
(163)
(164)
A−1 + B−1
(I + AB)−1
=
=
A(A + B)−1 B = B(A + B)−1 A
B − B(A + B)−1 B
(I + AB)−1 A
=
A(I + BA)−1
(167)
A−1 (A + B)B−1
I − A(I + BA)−1 B
(165)
(166)
Rank-1 update of inverse of inner product
Denote A = (XT X)−1 and that X is extended to include a new column vector
in the end X̃ = [X v]. Then [34]
"
#
T
vvT XAT
−AXT v
A + vAX
T v−vT XAXT v
T
T
T
T
−1
v v−v XAX v
(X̃ X̃)
=
1
−vT XAT
vT v−vT XAXT v
3.2.7
vT v−vT XAXT v
Rank-1 update of Moore-Penrose Inverse
The following is a rank-1 update for the Moore-Penrose pseudo-inverse of real
valued matrices and proof can be found in [18]. The matrix G is defined below:
(A + cdT )+ = A+ + G
(168)
Using the the notation
β
=
1 + d T A+ c
+
(169)
v
=
A c
(170)
n
w
=
=
(171)
(172)
m
=
(A+ )T d
(I − AA+ )c
(I − A+ A)T d
(173)
the solution is given as six different cases, depending on the entities ||w||,
||m||, and β. Please note, that for any (column) vector v it holds that v+ =
vT
vT (vT v)−1 = ||v||
2 . The solution is:
Case 1 of 6: If ||w|| =
6 0 and ||m|| =
6 0. Then
G
=
=
−vw+ − (m+ )T nT + β(m+ )T w+
1
β
1
vwT −
mnT +
mwT
−
||w||2
||m||2
||m||2 ||w||2
(174)
(175)
Case 2 of 6: If ||w|| = 0 and ||m|| =
6 0 and β = 0. Then
G
=
=
−vv+ A+ − (m+ )T nT
1
1
−
vvT A+ −
mnT
2
||v||
||m||2
(176)
(177)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 19
3.3
Implication on Inverses
3 INVERSES
Case 3 of 6: If ||w|| = 0 and β 6= 0. Then
1
β
G = mvT A+ −
β
||v||2 ||m||2 + |β|2
||v||2
m+v
β
T
||m||2 + T
(A ) v + n
β
(178)
Case 4 of 6: If ||w|| =
6 0 and ||m|| = 0 and β = 0. Then
G
=
=
−A+ nn+ − vw+
1
1
A+ nnT −
vwT
−
||n||2
||w||2
(179)
(180)
Case 5 of 6: If ||m|| = 0 and β 6= 0. Then
G=
1 +
β
A nwT −
β
||n||2 ||w||2 + |β|2
||w||2 +
A n+v
β
||n||2
w+n
β
T
(181)
Case 6 of 6: If ||w|| = 0 and ||m|| = 0 and β = 0. Then
G
=
=
3.3
−vv+ A+ − A+ nn+ + v+ A+ nvn+
1
1
v T A+ n
−
vvT A+ −
A+ nnT +
vnT
2
2
||v||
||n||
||v||2 ||n||2
(182)
(183)
Implication on Inverses
(A + B)−1 = A−1 + B−1
If
AB−1 A = BA−1 B
then
(184)
See [25].
3.3.1
A PosDef identity
Assume P, R to be positive definite and invertible, then
(P−1 + BT R−1 B)−1 BT R−1 = PBT (BPBT + R)−1
(185)
See [30].
3.4
Approximations
The following identity is known as the Neuman series of a matrix, which holds
when |λi | < 1 for all eigenvalues λi
(I − A)−1 =
which is equivalent to
(I + A)−1 =
∞
X
An
(186)
(−1)n An
(187)
n=0
∞
X
n=0
When |λi | < 1 for all eigenvalues λi , it holds that A → 0 for n → ∞, and the
following approximations holds
(I − A)−1
(I + A)−1
∼
=
∼
=
I + A + A2
I − A + A2
(188)
(189)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 20
3.5
Generalized Inverse
3 INVERSES
The following approximation is from [22] and holds when A large and symmetric
A − A(I + A)−1 A ∼
= I − A−1
(190)
If σ 2 is small compared to Q and M then
(Q + σ 2 M)−1 ∼
= Q−1 − σ 2 Q−1 MQ−1
(191)
Proof:
(Q + σ 2 M)−1
(QQ
−1
2
−1
−1
Q + σ MQ Q)
((I + σ 2 MQ−1 )Q)−1
=
(192)
=
=
(193)
(194)
Q−1 (I + σ 2 MQ−1 )−1
(195)
This can be rewritten using the Taylor expansion:
Q−1 (I + σ 2 MQ−1 )−1
Q
3.5
3.5.1
−1
2
(I − σ MQ
−1
2
+ (σ MQ
−1 2
) − ...)
=
∼
=
Q
−1
2
−σ Q
−1
MQ
−1
(196)
(197)
Generalized Inverse
Definition
A generalized inverse matrix of the matrix A is any matrix A− such that (see
[26])
AA− A = A
(198)
The matrix A− is not unique.
3.6
3.6.1
Pseudo Inverse
Definition
The pseudo inverse (or Moore-Penrose inverse) of a matrix A is the matrix A+
that fulfils
I
II
AA+ A = A
A+ AA+ = A+
III
AA+ symmetric
IV
A+ A symmetric
The matrix A+ is unique and does always exist. Note that in case of complex matrices, the symmetric condition is substituted by a condition of being
Hermitian.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 21
3.6
Pseudo Inverse
3.6.2
3 INVERSES
Properties
Assume A+ to be the pseudo-inverse of A, then (See [3] for some of them)
(A+ )+
(AT )+
=
=
A
(A+ )T
(199)
(200)
(AH )+
(A∗ )+
=
=
(A+ )H
(A+ )∗
(201)
(202)
(A+ A)AH
(A+ A)AT
AH
AT
(203)
(204)
(cA)+
A+
=
6
=
=
=
(1/c)A+
(AT A)+ AT
(205)
(206)
A+
=
AT (AAT )+
(207)
T
+
(A A)
(AAT )+
=
=
A (A )
(AT )+ A+
(208)
(209)
A+
A+
=
=
(AH A)+ AH
AH (AAH )+
(210)
(211)
(AH A)+
=
A+ (AH )+
(212)
H +
(AA )
(AB)+
=
=
H +
(A ) A
(A+ AB)+ (ABB+ )+
(213)
(214)
f (AH A) − f (0)I
=
A+ [f (AAH ) − f (0)I]A
H
f (AA ) − f (0)I
+
T +
+
H
A[f (A A) − f (0)I]A
=
+
(215)
(216)
where A ∈ Cn×m .
Assume A to have full rank, then
(AA+ )(AA+ )
=
AA+
(A A)(A A)
Tr(AA+ )
=
=
+
A A
rank(AA+ )
(See [26])
(218)
(219)
Tr(A+ A)
=
rank(A+ A)
(See [26])
(220)
=
(A+ AB)+ (ABB+ )+
+
+
(217)
For two matrices it hold that
(AB)+
(A ⊗ B)
3.6.3
+
=
+
A ⊗B
+
(221)
(222)
Construction
Assume that A has full rank, then
A n×n
A n×m
A n×m
Square
Broad
Tall
rank(A) = n
rank(A) = n
rank(A) = m
⇒
⇒
⇒
A+ = A−1
A+ = AT (AAT )−1
A+ = (AT A)−1 AT
The so-called ”broad version” is also known as right inverse and the ”tall version” as the left inverse.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 22
3.6
Pseudo Inverse
3 INVERSES
Assume A does not have full rank, i.e. A is n × m and rank(A) = r <
min(n, m). The pseudo inverse A+ can be constructed from the singular value
decomposition A = UDVT , by
T
A+ = Vr D−1
r Ur
(223)
where Ur , Dr , and Vr are the matrices with the degenerated rows and columns
deleted. A different way is this: There do always exist two matrices C n × r
and D r × m of rank r, such that A = CD. Using these matrices it holds that
A+ = DT (DDT )−1 (CT C)−1 CT
(224)
See [3].
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 23
4 COMPLEX MATRICES
4
Complex Matrices
The complex scalar product r = pq can be written as
ℜr
ℜp −ℑp
ℜq
=
ℑr
ℑp ℜp
ℑq
4.1
(225)
Complex Derivatives
In order to differentiate an expression f (z) with respect to a complex z, the
Cauchy-Riemann equations have to be satisfied ([7]):
∂ℜ(f (z))
∂ℑ(f (z))
df (z)
=
+i
dz
∂ℜz
∂ℜz
(226)
and
∂ℜ(f (z)) ∂ℑ(f (z))
df (z)
= −i
+
dz
∂ℑz
∂ℑz
or in a more compact form:
∂f (z)
∂f (z)
=i
.
∂ℑz
∂ℜz
(227)
(228)
A complex function that satisfies the Cauchy-Riemann equations for points in a
region R is said yo be analytic in this region R. In general, expressions involving
complex conjugate or conjugate transpose do not satisfy the Cauchy-Riemann
equations. In order to avoid this problem, a more generalized definition of
complex derivative is used ([24], [6]):
• Generalized Complex Derivative:
1 ∂f (z)
∂f (z)
df (z)
=
−i
.
dz
2 ∂ℜz
∂ℑz
(229)
• Conjugate Complex Derivative
df (z)
1 ∂f (z)
∂f (z)
.
=
+i
∗
dz
2 ∂ℜz
∂ℑz
(230)
The Generalized Complex Derivative equals the normal derivative, when f is an
analytic function. For a non-analytic function such as f (z) = z ∗ , the derivative
equals zero. The Conjugate Complex Derivative equals zero, when f is an
analytic function. The Conjugate Complex Derivative has e.g been used by [21]
when deriving a complex gradient.
Notice:
∂f (z)
∂f (z)
df (z)
6=
+i
.
(231)
dz
∂ℜz
∂ℑz
• Complex Gradient Vector: If f is a real function of a complex vector z,
then the complex gradient vector is given by ([14, p. 798])
∇f (z)
=
=
df (z)
dz∗
∂f (z)
∂f (z)
+i
.
∂ℜz
∂ℑz
2
(232)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 24
4.1
Complex Derivatives
4 COMPLEX MATRICES
• Complex Gradient Matrix: If f is a real function of a complex matrix Z,
then the complex gradient matrix is given by ([2])
∇f (Z)
=
=
df (Z)
dZ∗
∂f (Z)
∂f (Z)
+i
.
∂ℜZ
∂ℑZ
2
(233)
These expressions can be used for gradient descent algorithms.
4.1.1
The Chain Rule for complex numbers
The chain rule is a little more complicated when the function of a complex
u = f (x) is non-analytic. For a non-analytic function, the following chain rule
can be applied ([7])
∂g(u)
∂x
=
=
∂g ∂u
∂g ∂u∗
+
∂u ∂x ∂u∗ ∂x
∂g ∂u ∂g ∗ ∗ ∂u∗
+
∂u ∂x
∂u
∂x
(234)
Notice, if the function is analytic, the second term reduces to zero, and the function is reduced to the normal well-known chain rule. For the matrix derivative
of a scalar function g(U), the chain rule can be written the following way:
∗
T
T
Tr(( ∂g(U)
Tr(( ∂g(U)
∂g(U)
∂U ) ∂U)
∂U∗ ) ∂U )
=
+
.
∂X
∂X
∂X
4.1.2
(235)
Complex Derivatives of Traces
If the derivatives involve complex numbers, the conjugate transpose is often involved. The most useful way to show complex derivative is to show the derivative
with respect to the real and the imaginary part separately. An easy example is:
∂Tr(X∗ )
∂Tr(XH )
=
∂ℜX
∂ℜX
∗
∂Tr(X )
∂Tr(XH )
i
=i
∂ℑX
∂ℑX
=
I
(236)
=
I
(237)
Since the two results have the same sign, the conjugate complex derivative (230)
should be used.
∂Tr(XT )
∂Tr(X)
=
∂ℜX
∂ℜX
∂Tr(XT )
∂Tr(X)
=i
i
∂ℑX
∂ℑX
=
I
(238)
=
−I
(239)
Here, the two results have different signs, and the generalized complex derivative
(229) should be used. Hereby, it can be seen that (100) holds even if X is a
complex number.
∂Tr(AXH )
∂ℜX
∂Tr(AXH )
i
∂ℑX
=
A
(240)
=
A
(241)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 25
4.2
Higher order and non-linear derivatives
∂Tr(AX∗ )
∂ℜX
∂Tr(AX∗ )
i
∂ℑX
4 COMPLEX MATRICES
=
AT
(242)
=
AT
(243)
∂Tr(XH X)
∂Tr(XXH )
=
∂ℜX
∂ℜX
H
∂Tr(XH X)
∂Tr(XX )
=i
i
∂ℑX
∂ℑX
=
2ℜX
(244)
=
i2ℑX
(245)
By inserting (244) and (245) in (229) and (230), it can be seen that
∂Tr(XXH )
= X∗
∂X
∂Tr(XXH )
=X
∂X∗
(246)
(247)
Since the function Tr(XXH ) is a real function of the complex matrix X, the
complex gradient matrix (233) is given by
∇Tr(XXH ) = 2
4.1.3
∂Tr(XXH )
= 2X
∂X∗
(248)
Complex Derivative Involving Determinants
Here, a calculation example is provided. The objective is to find the derivative of
det(XH AX) with respect to X ∈ Cm×n . The derivative is found with respect to
the real part and the imaginary part of X, by use of (42) and (37), det(XH AX)
can be calculated as (see App. B.1.4 for details)
∂ det(XH AX)
∂X
=
=
1 ∂ det(XH AX)
∂ det(XH AX)
−i
2
∂ℜX
∂ℑX
T
det(XH AX) (XH AX)−1 XH A
(249)
and the complex conjugate derivative yields
∂ det(XH AX)
∂X∗
=
=
4.2
1 ∂ det(XH AX)
∂ det(XH AX)
+i
2
∂ℜX
∂ℑX
det(XH AX)AX(XH AX)−1
(250)
Higher order and non-linear derivatives
∂ (Ax)H (Ax)
∂x (Bx)H (Bx)
=
=
∂ xH AH Ax
∂x xH BH Bx
AH Ax
xH AH AxBH Bx
2 H
−2
x BBx
(xH BH Bx)2
(251)
(252)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 26
4.3
4.3
Inverse of complex sum
4 COMPLEX MATRICES
Inverse of complex sum
Given real matrices A, B find the inverse of the complex sum A + iB. Form
the auxiliary matrices
E
=
A + tB
(253)
F
=
B − tA,
(254)
and find a value of t such that E−1 exists. Then
(A + iB)−1
=
=
=
=
=
(1 − it)(E + iF)−1
(255)
−1
−1
−1
−1
−1
(1 − it)((E + FE F) − i(E + FE F) FE )(256)
(1 − it)(E + FE−1 F)−1 (I − iFE−1 )
(E + FE−1 F)−1 ((I − tFE−1 ) − i(tI + FE−1 ))
(257)
(258)
−i(E + FE−1 F)−1 (tI + FE−1 )
(259)
(E + FE−1 F)−1 (I − tFE−1 )
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 27
5 SOLUTIONS AND DECOMPOSITIONS
5
5.1
5.1.1
Solutions and Decompositions
Solutions to linear equations
Simple Linear Regression
Assume we have data (xn , yn ) for n = 1, ..., N and are seeking the parameters
a, b ∈ R such that yi ∼
= axi + b. With a least squares error function, the optimal
values for a, b can be expressed using the notation
x = (x1 , ..., xN )T
y = (y1 , ..., yN )T
1 = (1, ..., 1)T ∈ RN ×1
and
Rxx = xT x
Ryx = yT x
Rx1 = xT 1 R11 = 1T 1
Ry1 = yT 1
as
5.1.2
a
b
=
Rxx
Rx1
Rx1
R11
−1
Rx,y
Ry1
(260)
Existence in Linear Systems
Assume A is n × m and consider the linear system
Ax = b
(261)
Construct the augmented matrix B = [A b] then
Condition
rank(A) = rank(B) = m
rank(A) = rank(B) < m
rank(A) < rank(B)
5.1.3
Solution
Unique solution x
Many solutions x
No solutions x
Standard Square
Assume A is square and invertible, then
Ax = b
5.1.4
⇒
x = A−1 b
(262)
Degenerated Square
Assume A is n × n but of rank r < n. In that case, the system Ax = b is solved
by
x = A+ b
where A+ is the pseudo-inverse of the rank-deficient matrix, constructed as
described in section 3.6.3.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 28
5.1
Solutions to linear equations5 SOLUTIONS AND DECOMPOSITIONS
5.1.5
Cramer’s rule
The equation
Ax = b,
(263)
where A is square has exactly one solution x if the ith element in x can be
found as
det B
xi =
,
(264)
det A
where B equals A, but the ith column in A has been substituted by b.
5.1.6
Over-determined Rectangular
Assume A to be n × m, n > m (tall) and rank(A) = m, then
Ax = b
x = (AT A)−1 AT b = A+ b
⇒
(265)
that is if there exists a solution x at all! If there is no solution the following
can be useful:
Ax = b
⇒
xmin = A+ b
(266)
Now xmin is the vector x which minimizes ||Ax − b||2 , i.e. the vector which is
”least wrong”. The matrix A+ is the pseudo-inverse of A. See [3].
5.1.7
Under-determined Rectangular
Assume A is n × m and n < m (”broad”) and rank(A) = n.
Ax = b
xmin = AT (AAT )−1 b
⇒
(267)
The equation have many solutions x. But xmin is the solution which minimizes
||Ax − b||2 and also the solution with the smallest norm ||x||2 . The same holds
for a matrix version: Assume A is n × m, X is m × n and B is n × n, then
⇒
AX = B
Xmin = A+ B
(268)
The equation have many solutions X. But Xmin is the solution which minimizes
||AX − B||2 and also the solution with the smallest norm ||X||2 . See [3].
Similar but different: Assume A is square n × n and the matrices B0 , B1
are n × N , where N > n, then if B0 has maximal rank
AB0 = B1
Amin = B1 BT0 (B0 BT0 )−1
⇒
(269)
where Amin denotes the matrix which is optimal in a least square sense. An
interpretation is that A is the linear approximation which maps the columns
vectors of B0 into the columns vectors of B1 .
5.1.8
Linear form and zeros
Ax = 0,
5.1.9
∀x
⇒
A=0
(270)
Square form and zeros
If A is symmetric, then
xT Ax = 0,
∀x
⇒
A=0
(271)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 29
5.2
Eigenvalues and Eigenvectors5 SOLUTIONS AND DECOMPOSITIONS
5.1.10
The Lyapunov Equation
AX + XB
=
C
(272)
vec(X)
=
(I ⊗ A + BT ⊗ I)−1 vec(C)
(273)
Sec 10.2.1 and 10.2.2 for details on the Kronecker product and the vec operator.
5.1.11
Encapsulating Sum
P
n An XBn
vec(X)
=
C
P
=
(274)
T
n Bn
⊗ An
−1
vec(C)
(275)
See Sec 10.2.1 and 10.2.2 for details on the Kronecker product and the vec
operator.
5.2
5.2.1
Eigenvalues and Eigenvectors
Definition
The eigenvectors vi and eigenvalues λi are the ones satisfying
Avi = λi vi
5.2.2
(276)
Decompositions
For matrices A with as many distinct eigenvalues as dimensions, the following
holds, where the columns of V are the eigenvectors and (D)ij = δij λi ,
AV = VD
(277)
For defective matrices A, which is matrices which has fewer distinct eigenvalues
than dimensions, the following decomposition called Jordan canonical form,
holds
AV = VJ
(278)
where J is a block diagonal matrix with the blocks Ji = λi I + N. The matrices
Ji have dimensionality as the number of identical eigenvalues equal to λi , and N
is square matrix of same size with 1 on the super diagonal and zero elsewhere.
It also holds that for all matrices A there exists matrices V and R such that
AV = VR
(279)
where R is upper triangular with the eigenvalues λi on its diagonal.
5.2.3
General Properties
Assume that A ∈ Rn×m and B ∈ Rm×n ,
eig(AB)
=
eig(BA)
(280)
rank(A) = r
⇒
At most r non-zero λi
(281)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 30
5.3
Singular Value Decomposition
5 SOLUTIONS AND DECOMPOSITIONS
5.2.4
Symmetric
Assume A is symmetric, then
VVT
λi
=
∈
Tr(Ap )
I
R
P
=
eig(I + cA)
eig(A − cI)
=
=
eig(A−1 )
(i.e. V is orthogonal)
(i.e. λi is real)
p
i λi
(284)
1 + cλi
λi − c
(285)
(286)
λ−1
i
=
(282)
(283)
(287)
For a symmetric, positive matrix A,
eig(AT A) = eig(AAT ) = eig(A) ◦ eig(A)
5.2.5
(288)
Characteristic polynomial
The characteristic polynomial for the matrix A is
0 =
=
det(A − λI)
n
λ − g1 λ
n−1
(289)
+ g2 λ
n−2
n
− ... + (−1) gn
(290)
Note that the coefficients gj for j = 1, ..., n are the n invariants under rotation
of A. Thus, gj is the sum of the determinants of all the sub-matrices of A taken
j rows and columns at a time. That is, g1 is the trace of A, and g2 is the sum
of the determinants of the n(n − 1)/2 sub-matrices that can be formed from A
by deleting all but two rows and columns, and so on – see [17].
5.3
Singular Value Decomposition
Any n × m matrix A can be written as
A = UDVT ,
where
5.3.1
U
D
V
=
=
=
(291)
eigenvectors of AAT
p
diag(eig(AAT ))
eigenvectors of AT A
n×n
n×m
m×m
(292)
Symmetric Square decomposed into squares
Assume A to be n × n and symmetric. Then
T
A = V
D
V
,
(293)
where D is diagonal with the eigenvalues of A, and V is orthogonal and the
eigenvectors of A.
5.3.2
Square decomposed into squares
Assume A ∈ Rn×n . Then
A
=
V
D
UT
,
(294)
where D is diagonal with the square root of the eigenvalues of AAT , V is the
eigenvectors of AAT and UT is the eigenvectors of AT A.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 31
5.4
Triangular Decomposition
5.3.3
5 SOLUTIONS AND DECOMPOSITIONS
Square decomposed into rectangular
Assume V∗ D∗ UT∗ = 0 then we can expand the SVD of A into
T
D 0
U
A = V V∗
,
0 D∗
UT∗
(295)
where the SVD of A is A = VDUT .
5.3.4
Rectangular decomposition I
Assume A is n × m, V is n × n, D is n × n, UT is n × m
A
D
UT
,
= V
(296)
where D is diagonal with the square root of the eigenvalues of AAT , V is the
eigenvectors of AAT and UT is the eigenvectors of AT A.
5.3.5
Rectangular decomposition II
Assume A is n × m, V is n × m, D is m × m, UT is m × m
A
V
D
UT
=
5.3.6
(297)
Rectangular decomposition III
Assume A is n × m, V is n × n, D is n × m, UT is m × m
,
A
D
UT
= V
(298)
where D is diagonal with the square root of the eigenvalues of AAT , V is the
eigenvectors of AAT and UT is the eigenvectors of AT A.
5.4
Triangular Decomposition
5.5
LU decomposition
Assume A is a square matrix with non-zero leading principal minors, then
A = LU
(299)
where L is a unique unit lower triangular matrix and U is a unique upper
triangular matrix.
5.5.1
Cholesky-decomposition
Assume A is a symmetric positive definite square matrix, then
A = UT U = LLT ,
(300)
where U is a unique upper triangular matrix and L is a lower triangular matrix.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 32
5.6
5.6
LDM decomposition
5 SOLUTIONS AND DECOMPOSITIONS
LDM decomposition
Assume A is a square matrix with non-zero leading principal minors1 , then
A = LDMT
(301)
where L, M are unique unit lower triangular matrices and D is a unique diagonal
matrix.
5.7
LDL decompositions
The LDL decomposition are special cases of the LDM decomposition. Assume
A is a non-singular symmetric definite square matrix, then
A = LDLT = LT DL
(302)
where L is a unit lower triangular matrix and D is a diagonal matrix. If A is
also positive definite, then D has strictly positive diagonal entries.
1 If the matrix that corresponds to a principal minor is a quadratic upper-left part of the
larger matrix (i.e., it consists of matrix elements in rows and columns from 1 to k), then the
principal minor is called a leading principal minor. For an n times n square matrix, there are
n leading principal minors. [31]
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 33
6 STATISTICS AND PROBABILITY
6
Statistics and Probability
6.1
Definition of Moments
Assume x ∈ Rn×1 is a random variable
6.1.1
Mean
The vector of means, m, is defined by
(m)i = hxi i
6.1.2
(303)
Covariance
The matrix of covariance M is defined by
(M)ij = h(xi − hxi i)(xj − hxj i)i
(304)
M = h(x − m)(x − m)T i
(305)
or alternatively as
6.1.3
Third moments
The matrix of third centralized moments – in some contexts referred to as
coskewness – is defined using the notation
(3)
mijk = h(xi − hxi i)(xj − hxj i)(xk − hxk i)i
(306)
i
h
(3) (3)
M3 = m::1 m::2 ...m(3)
::n
(307)
as
where ’:’ denotes all elements within the given index. M3 can alternatively be
expressed as
M3 = h(x − m)(x − m)T ⊗ (x − m)T i
(308)
6.1.4
Fourth moments
The matrix of fourth centralized moments – in some contexts referred to as
cokurtosis – is defined using the notation
(4)
mijkl = h(xi − hxi i)(xj − hxj i)(xk − hxk i)(xl − hxl i)i
(309)
as
i
h
(4)
(4)
(4)
(4)
(4)
(4)
(4)
(4)
M4 = m::11 m::21 ...m::n1 |m::12 m::22 ...m::n2 |...|m::1n m::2n ...m(4)
::nn
(310)
or alternatively as
M4 = h(x − m)(x − m)T ⊗ (x − m)T ⊗ (x − m)T i
(311)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 34
6.2
Expectation of Linear Combinations
6 STATISTICS AND PROBABILITY
6.2
Expectation of Linear Combinations
6.2.1
Linear Forms
Assume X and x to be a matrix and a vector of random variables. Then (see
See [26])
E[AXB + C]
Var[Ax]
=
=
AE[X]B + C
AVar[x]AT
(312)
(313)
Cov[Ax, By]
=
ACov[x, y]BT
(314)
Assume x to be a stochastic vector with mean m, then (see [7])
6.2.2
E[Ax + b]
=
Am + b
(315)
E[Ax]
E[x + b]
=
=
Am
m+b
(316)
(317)
Quadratic Forms
Assume A is symmetric, c = E[x] and Σ = Var[x]. Assume also that all
coordinates xi are independent, have the same central moments µ1 , µ2 , µ3 , µ4
and denote a = diag(A). Then (See [26])
E[xT Ax]
=
Tr(AΣ) + cT Ac
T
=
2µ22 Tr(A2 ) + 4µ2 cT A2 c + 4µ3 cT Aa + (µ4 − 3µ22 )aT a (319)
Var[x Ax]
(318)
Also, assume x to be a stochastic vector with mean m, and covariance M. Then
(see [7])
E[(Ax + a)(Bx + b)T ]
E[xxT ]
E[xaT x]
=
=
=
AMBT + (Am + a)(Bm + b)T
M + mmT
(M + mmT )a
(320)
(321)
(322)
E[xT axT ]
=
aT (M + mmT )
(323)
T
E[(Ax)(Ax) ]
E[(x + a)(x + a)T ]
=
=
T
T
A(M + mm )A
M + (m + a)(m + a)T
(324)
(325)
E[(Ax + a)T (Bx + b)]
E[xT x]
=
=
Tr(AMBT ) + (Am + a)T (Bm + b) (326)
Tr(M) + mT m
(327)
E[xT Ax]
=
Tr(AM) + mT Am
T
E[(Ax) (Ax)]
E[(x + a)T (x + a)]
=
=
T
(328)
T
Tr(AMA ) + (Am) (Am)
Tr(M) + (m + a)T (m + a)
(329)
(330)
See [7].
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 35
6.3
Weighted Scalar Variable
6.2.3
6 STATISTICS AND PROBABILITY
Cubic Forms
Assume x to be a stochastic vector with independent coordinates, mean m,
covariance M and central moments v3 = E[(x − m)3 ]. Then (see [7])
E[(Ax + a)(Bx + b)T (Cx + c)]
Adiag(BT C)v3
=
+Tr(BMCT )(Am + a)
+AMCT (Bm + b)
E[xx x]
=
+(AMBT + (Am + a)(Bm + b)T )(Cm + c)
v3 + 2Mm + (Tr(M) + mT m)m
E[(Ax + a)(Ax + a) (Ax + a)]
=
Adiag(AT A)v3
T
T
+[2AMAT + (Ax + a)(Ax + a)T ](Am + a)
+Tr(AMAT )(Am + a)
E[(Ax + a)bT (Cx + c)(Dx + d)T ]
(Ax + a)bT (CMDT + (Cm + c)(Dm + d)T )
+(AMCT + (Am + a)(Cm + c)T )b(Dm + d)T
=
+bT (Cm + c)(AMDT − (Am + a)(Dm + d)T )
6.3
Weighted Scalar Variable
Assume x ∈ Rn×1 is a random variable, w ∈ Rn×1 is a vector of constants and
y is the linear combination y = wT x. Assume further that m, M2 , M3 , M4
denotes the mean, covariance, and central third and fourth moment matrix of
the variable x. Then it holds that
hyi
2
h(y − hyi) i
h(y − hyi)3 i
h(y − hyi)4 i
=
wT m
(331)
=
=
T
(332)
(333)
=
w M2 w
w T M3 w ⊗ w
w T M4 w ⊗ w ⊗ w
(334)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 36
7 MULTIVARIATE DISTRIBUTIONS
7
Multivariate Distributions
7.1
Cauchy
The density function for a Cauchy distributed vector t ∈ RP ×1 , is given by
p(t|µ, Σ) = π −P/2
Γ( 1+P
det(Σ)−1/2
2 )
Γ(1/2) 1 + (t − µ)T Σ−1 (t − µ)(1+P )/2
(335)
where µ is the location, Σ is positive definite, and Γ denotes the gamma function. The Cauchy distribution is a special case of the Student-t distribution.
7.2
Dirichlet
The Dirichlet distribution is a kind of “inverse” distribution compared to the
multinomial distribution on the bounded continuous variate x = [x1 , . . . , xP ]
[16, p. 44]
P
P
P
Γ
p αp Y α −1
xp p
p(x|α) = QP
Γ(α
)
p
p
p
7.3
Normal
The normal distribution is also known as a Gaussian distribution. See sec. 8.
7.4
Normal-Inverse Gamma
7.5
Gaussian
See sec. 8.
7.6
Multinomial
If the vector n contains counts, i.e. (n)i ∈ 0, 1, 2, ..., then the discrete multinomial disitrbution for n is given by
d
P (n|a, n) =
where ai are probabilities, i.e. 0 ≤ ai ≤ 1 and
7.7
Student’s t
d
X
Y
n!
a ni ,
n1 ! . . . nd ! i i
P
ni = n
(336)
i
i
ai = 1.
The density of a Student-t distributed vector t ∈ RP ×1 , is given by
p(t|µ, Σ, ν) = (πν)−P/2
Γ( ν+P
det(Σ)−1/2
2 )
Γ(ν/2) 1 + ν −1 (t − µ)T Σ−1 (t − µ)(ν+P )/2
(337)
where µ is the location, the scale matrix Σ is symmetric, positive definite, ν
is the degrees of freedom, and Γ denotes the gamma function. For ν = 1, the
Student-t distribution becomes the Cauchy distribution (see sec 7.1).
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 37
7.8
Wishart
7.7.1
7 MULTIVARIATE DISTRIBUTIONS
Mean
E(t) = µ,
7.7.2
(338)
Variance
cov(t) =
7.7.3
ν>1
ν
Σ,
ν−2
ν>2
(339)
Mode
The notion mode meaning the position of the most probable value
mode(t) = µ
7.7.4
(340)
Full Matrix Version
If instead of a vector t ∈ RP ×1 one has a matrix T ∈ RP ×N , then the Student-t
distribution for T is
p(T|M, Ω, Σ, ν)
=
π −N P/2
P
Y
Γ [(ν + P − p + 1)/2]
×
Γ [(ν − p + 1)/2]
p=1
ν det(Ω)−ν/2 det(Σ)−N/2 ×
−(ν+P )/2
det Ω−1 + (T − M)Σ−1 (T − M)T
(341)
where M is the location, Ω is the rescaling matrix, Σ is positive definite, ν is
the degrees of freedom, and Γ denotes the gamma function.
7.8
Wishart
The central Wishart distribution for M ∈ RP ×P , M is positive definite, where
m can be regarded as a degree of freedom parameter [16, equation 3.8.1] [8,
section 2.5],[11]
p(M|Σ, m)
7.8.1
=
2mP/2 π P (P −1)/4
1
QP
p
Γ[ 12 (m + 1 − p)]
×
det(Σ)−m/2 det(M)(m−P −1)/2 ×
1
−1
exp − Tr(Σ M)
2
(342)
E(M) = mΣ
(343)
Mean
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 38
7.9
Wishart, Inverse
7.9
7 MULTIVARIATE DISTRIBUTIONS
Wishart, Inverse
The (normal) Inverse Wishart distribution for M ∈ RP ×P , M is positive definite, where m can be regarded as a degree of freedom parameter [11]
p(M|Σ, m)
=
2mP/2 π P (P −1)/4
1
QP
p
Γ[ 12 (m + 1 − p)]
det(Σ)m/2 det(M)−(m−P −1)/2 ×
1
exp − Tr(ΣM−1 )
2
7.9.1
×
(344)
Mean
E(M) = Σ
1
m−P −1
(345)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 39
8 GAUSSIANS
8
8.1
8.1.1
Gaussians
Basics
Density and normalization
The density of x ∼ N (m, Σ) is
1
1
T −1
p
p(x) =
exp − (x − m) Σ (x − m)
2
det(2πΣ)
Note that if x is d-dimensional, then
Integration and normalization
Z
1
exp − (x − m)T Σ−1 (x − m) dx
2
Z
1
exp − xT Σ−1 x + mT Σ−1 x dx
2
Z
1 T
T
exp − x Ax + c x dx
2
(346)
det(2πΣ) = (2π)d det(Σ).
=
=
=
p
det(2πΣ)
p
1
det(2πΣ) exp mT Σ−1 m
2
p
1 T −T
−1
det(2πA ) exp c A c
2
If X = [x1 x2 ...xn ] and C = [c1 c2 ...cn ], then
Z
p
n
1
1
T
T
T −1
−1
exp − Tr(X AX) + Tr(C X) dX = det(2πA ) exp Tr(C A C)
2
2
The derivatives of the density are
∂p(x)
∂x
∂2p
∂x∂xT
8.1.2
=
=
−p(x)Σ−1 (x − m)
p(x) Σ−1 (x − m)(x − m)T Σ−1 − Σ−1
(347)
(348)
Marginal Distribution
Assume x ∼ Nx (µ, Σ) where
xa
µa
µ=
x=
xb
µb
Σ=
Σa
ΣTc
Σc
Σb
(349)
then
p(xa )
p(xb )
8.1.3
=
=
Nxa (µa , Σa )
Nxb (µb , Σb )
(350)
(351)
Conditional Distribution
Assume x ∼ Nx (µ, Σ) where
xa
µa
x=
µ=
xb
µb
Σ=
Σa
ΣTc
Σc
Σb
(352)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 40
8.1
Basics
8 GAUSSIANS
then
n µ̂
= µa + Σc Σ−1
a
b (xb − µb ) (353)
T
Σ̂a = Σa − Σc Σ−1
b Σc
n µ̂ = µ + ΣT Σ−1 (x − µ )
a
b
b
a
c
a
(354)
Σ̂b = Σb − ΣTc Σ−1
a Σc
p(xa |xb ) = Nxa (µ̂a , Σ̂a )
p(xb |xa ) = Nxb (µ̂b , Σ̂b )
Note, that the covariance matrices are the Schur complement of the block matrix, see 9.1.5 for details.
8.1.4
Linear combination
Assume x ∼ N (mx , Σx ) and y ∼ N (my , Σy ) then
Ax + By + c ∼ N (Amx + Bmy + c, AΣx AT + BΣy BT )
(355)
Rearranging Means
p
det(2π(AT Σ−1 A)−1 )
p
NAx [m, Σ] =
Nx [A−1 m, (AT Σ−1 A)−1 ]
det(2πΣ)
(356)
8.1.5
If A is square and invertible, it simplifies to
NAx [m, Σ] =
8.1.6
1
Nx [A−1 m, (AT Σ−1 A)−1 ]
| det(A)|
(357)
Rearranging into squared form
If A is symmetric, then
1
− xT Ax + bT x
2
=
1
− Tr(XT AX) + Tr(BT X)
2
=
8.1.7
1
1
− (x − A−1 b)T A(x − A−1 b) + bT A−1 b
2
2
1
1
− Tr[(X − A−1 B)T A(X − A−1 B)] + Tr(BT A−1 B)
2
2
Sum of two squared forms
In vector formulation (assuming Σ1 , Σ2 are symmetric)
=
Σ−1
c
mc
=
=
C
=
1
− (x − m1 )T Σ−1
1 (x − m1 )
2
1
− (x − m2 )T Σ−1
2 (x − m2 )
2
1
− (x − mc )T Σ−1
c (x − mc ) + C
2
(358)
(359)
(360)
−1
Σ−1
(361)
1 + Σ2
−1
−1 −1
−1
−1
(Σ1 + Σ2 ) (Σ1 m1 + Σ2 m2 )
(362)
1 T −1
−1
−1 −1
−1
(m Σ + mT2 Σ−1
(Σ−1
2 )(Σ1 + Σ2 )
1 m1 + Σ2 m2 )(363)
2 1 1
1
T −1
(364)
− mT1 Σ−1
1 m1 + m2 Σ 2 m2
2
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 41
8.2
Moments
8 GAUSSIANS
In a trace formulation (assuming Σ1 , Σ2 are symmetric)
=
Σ−1
c
Mc
=
=
C
=
8.1.8
1
− Tr((X − M1 )T Σ−1
1 (X − M1 ))
2
1
− Tr((X − M2 )T Σ−1
2 (X − M2 ))
2
1
− Tr[(X − Mc )T Σ−1
c (X − Mc )] + C
2
(365)
(366)
(367)
−1
Σ−1
(368)
1 + Σ2
−1
−1 −1
−1
−1
(Σ1 + Σ2 ) (Σ1 M1 + Σ2 M2 )
(369)
i
1 h −1
−1
−1 −1
−1
−1
T
Tr (Σ1 M1 + Σ−1
M
)
(Σ
+
Σ
)
(Σ
M
+
Σ
M
)
2
1
2
2
1
2
1
2
2
1
T −1
(370)
− Tr(MT1 Σ−1
1 M 1 + M 2 Σ2 M 2 )
2
Product of gaussian densities
Let Nx (m, Σ) denote a density of x, then
Nx (m1 , Σ1 ) · Nx (m2 , Σ2 ) = cc Nx (mc , Σc )
cc
=
=
mc
Σc
=
=
(371)
Nm1 (m2 , (Σ1 + Σ2 ))
1
1
p
exp − (m1 − m2 )T (Σ1 + Σ2 )−1 (m1 − m2 )
2
det(2π(Σ1 + Σ2 ))
−1 −1
−1
(Σ−1
(Σ−1
1 + Σ2 )
1 m1 + Σ2 m2 )
−1 −1
(Σ−1
1 + Σ2 )
but note that the product is not normalized as a density of x.
8.2
8.2.1
Moments
Mean and covariance of linear forms
First and second moments. Assume x ∼ N (m, Σ)
E(x) = m
(372)
Cov(x, x) = Var(x) = Σ = E(xxT ) − E(x)E(xT ) = E(xxT ) − mmT
(373)
As for any other distribution is holds for gaussians that
E[Ax]
Var[Ax]
Cov[Ax, By]
=
=
=
AE[x]
(374)
T
AVar[x]A
ACov[x, y]BT
(375)
(376)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 42
8.2
Moments
8.2.2
8 GAUSSIANS
Mean and variance of square forms
Mean and variance of square forms: Assume x ∼ N (m, Σ)
E(xxT )
E[xT Ax]
=
=
Σ + mmT
Tr(AΣ) + mT Am
(377)
(378)
Var(xT Ax)
=
Tr[AΣ(A + AT )Σ] + ...
+mT (A + AT )Σ(A + AT )m
(379)
E[(x − m′ )T A(x − m′ )]
=
(m − m′ )T A(m − m′ ) + Tr(AΣ)
(380)
If Σ = σ 2 I and A is symmetric, then
Var(xT Ax)
=
2σ 4 Tr(A2 ) + 4σ 2 mT A2 m
(381)
Assume x ∼ N (0, σ 2 I) and A and B to be symmetric, then
Cov(xT Ax, xT Bx) = 2σ 4 Tr(AB)
8.2.3
(382)
Cubic forms
Assume x to be a stochastic vector with independent coordinates, mean m and
covariance M
E[xbT xxT ]
=
mbT (M + mmT ) + (M + mmT )bmT
+bT m(M − mmT )
8.2.4
(383)
Mean of Quartic Forms
E[xxT xxT ]
T
T
E[xx Axx ]
E[xT xxT x]
E[xT AxxT Bx]
=
=
=
=
2(Σ + mmT )2 + mT m(Σ − mmT )
+Tr(Σ)(Σ + mmT )
(Σ + mmT )(A + AT )(Σ + mmT )
+mT Am(Σ − mmT ) + Tr[AΣ](Σ + mmT )
2Tr(Σ2 ) + 4mT Σm + (Tr(Σ) + mT m)2
Tr[AΣ(B + BT )Σ] + mT (A + AT )Σ(B + BT )m
+(Tr(AΣ) + mT Am)(Tr(BΣ) + mT Bm)
=
E[aT xbT xcT xdT x]
(aT (Σ + mmT )b)(cT (Σ + mmT )d)
+(aT (Σ + mmT )c)(bT (Σ + mmT )d)
+(aT (Σ + mmT )d)(bT (Σ + mmT )c) − 2aT mbT mcT mdT m
E[(Ax + a)(Bx + b)T (Cx + c)(Dx + d)T ]
=
[AΣBT + (Am + a)(Bm + b)T ][CΣDT + (Cm + c)(Dm + d)T ]
+[AΣCT + (Am + a)(Cm + c)T ][BΣDT + (Bm + b)(Dm + d)T ]
+(Bm + b)T (Cm + c)[AΣDT − (Am + a)(Dm + d)T ]
+Tr(BΣCT )[AΣDT + (Am + a)(Dm + d)T ]
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 43
8.3
Miscellaneous
=
8 GAUSSIANS
E[(Ax + a)T (Bx + b)(Cx + c)T (Dx + d)]
Tr[AΣ(CT D + DT C)ΣBT ]
+[(Am + a)T B + (Bm + b)T A]Σ[CT (Dm + d) + DT (Cm + c)]
+[Tr(AΣBT ) + (Am + a)T (Bm + b)][Tr(CΣDT ) + (Cm + c)T (Dm + d)]
See [7].
8.2.5
Moments
E[x]
X
=
ρk m k
(384)
k
Cov(x)
XX
=
k
8.3
8.3.1
k′
ρk ρk′ (Σk + mk mTk − mk mTk′ )
(385)
Miscellaneous
Whitening
Assume x ∼ N (m, Σ) then
z = Σ−1/2 (x − m) ∼ N (0, I)
(386)
Conversely having z ∼ N (0, I) one can generate data x ∼ N (m, Σ) by setting
x = Σ1/2 z + m ∼ N (m, Σ)
(387)
Note that Σ1/2 means the matrix which fulfils Σ1/2 Σ1/2 = Σ, and that it exists
and is unique since Σ is positive definite.
8.3.2
The Chi-Square connection
Assume x ∼ N (m, Σ) and x to be n dimensional, then
z = (x − m)T Σ−1 (x − m) ∼ χ2n
(388)
where χ2n denotes the Chi square distribution with n degrees of freedom.
8.3.3
Entropy
Entropy of a D-dimensional gaussian
Z
p
D
H(x) = − N (m, Σ) ln N (m, Σ)dx = ln det(2πΣ) +
2
8.4
8.4.1
(389)
Mixture of Gaussians
Density
The variable x is distributed as a mixture of gaussians if it has the density
K
X
1
exp − (x − mk )T Σ−1
p(x) =
ρk p
k (x − mk )
2
det(2πΣk )
k=1
1
(390)
where ρk sum to 1 and the Σk all are positive definite.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 44
8.4
Mixture of Gaussians
8 GAUSSIANS
8.4.2
Derivatives
P
Defining p(s) = k ρk Ns (µk , Σk ) one get
∂ ln p(s)
∂ρj
=
=
∂ ln p(s)
∂µj
=
=
∂ ln p(s)
∂Σj
=
=
ρj Ns (µj , Σj )
∂
P
ln[ρj Ns (µj , Σj )]
(391)
ρ
N
(µ
,
Σ
)
∂ρ
k
s
k
j
k
k
ρj Ns (µj , Σj ) 1
P
(392)
k ρk Ns (µk , Σk ) ρj
ρj Ns (µj , Σj )
∂
P
ln[ρj Ns (µj , Σj )]
(393)
ρ
N
(µ
,
Σ
)
∂µ
k
k
j
k k s
ρj Ns (µj , Σj ) −1
P
Σj (s − µj )
(394)
k ρk Ns (µk , Σk )
ρj Ns (µj , Σj )
∂
P
ln[ρj Ns (µj , Σj )]
(395)
ρ
N
(µ
,
Σ
)
∂Σ
k
j
k
k k s
ρj Ns (µj , Σj ) 1
−1
T −1
P
−Σ−1
j + Σj (s − µj )(s − µj ) Σj (396)
ρ
N
(µ
,
Σ
)
2
k
k
k k s
But ρk and Σk needs to be constrained.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 45
9 SPECIAL MATRICES
9
9.1
Special Matrices
Block matrices
Let Aij denote the ijth block of A.
9.1.1
Multiplication
Assuming the dimensions of the blocks matches we have
A11 A12
B11 B12
A11 B11 + A12 B21 A11 B12 + A12 B22
=
A21 A22
B21 B22
A21 B11 + A22 B21 A21 B12 + A22 B22
9.1.2
The Determinant
The determinant can be expressed as by the use of
C1
C2
as
det
9.1.3
A12
A22
A11
A21
A11 − A12 A−1
22 A21
=
A22 −
=
(397)
A21 A−1
11 A12
(398)
= det(A22 ) · det(C1 ) = det(A11 ) · det(C2 )
The Inverse
The inverse can be expressed as by the use of
C1
as
=
9.1.4
A12
A22
A11
A21
A11 − A12 A−1
22 A21
=
A22 −
C2
=
−1
=
C−1
1
−1
−C2 A21 A−1
11
−1
−1
−1
A−1
11 + A11 A12 C2 A21 A11
−1
−1
−A22 A21 C1
(399)
A21 A−1
11 A12
(400)
−1
−A−1
11 A12 C2
−1
C2
−1
−C−1
1 A12 A22
−1
−1
−1
A22 + A22 A21 C1 A12 A−1
22
Block diagonal
For block diagonal matrices we have
det
A11
0
A11
0
0
A22
0
A22
−1
(A11 )−1
0
=
=
det(A11 ) · det(A22 )
0
(A22 )−1
(401)
(402)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 46
9.2
Discrete Fourier Transform Matrix, The
9.1.5
9 SPECIAL MATRICES
Schur complement
Regard the matrix
A11
A21
A12
A22
The Schur complement of block A11 of the matrix above is the matrix (denoted
C2 in the text above)
A22 − A21 A−1
11 A12
The Schur complement of block A22 of the matrix above is the matrix (denoted
C1 in the text above)
A11 − A12 A−1
22 A21
Using the Schur complement, one can rewrite the inverse of a block matrix
=
A12
A22
A11
A21
I
−A−1
22 A21
−1
0
I
−1
(A11 − A12 A−1
22 A21 )
0
0
A−1
22
I
0
−A12 A−1
22
I
The Schur complement is useful when solving linear systems of the form
A11 A12
x1
b1
=
A21 A22
x2
b2
which has the following equation for x1
−1
(A11 − A12 A−1
22 A21 )x1 = b1 − A12 A22 b2
When the appropriate inverses exists, this can be solved for x1 which can then
be inserted in the equation for x2 to solve for x2 .
9.2
Discrete Fourier Transform Matrix, The
The DFT matrix is an N × N symmetric matrix WN , where the k, nth element
is given by
−j2πkn
(403)
WNkn = e N
Thus the discrete Fourier transform (DFT) can be expressed as
X(k) =
N
−1
X
x(n)WNkn .
(404)
n=0
Likewise the inverse discrete Fourier transform (IDFT) can be expressed as
x(n) =
N −1
1 X
X(k)WN−kn .
N
(405)
k=0
The DFT of the vector x = [x(0), x(1), · · · , x(N − 1)]T can be written in matrix
form as
X = WN x,
(406)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 47
9.3
Hermitian Matrices and skew-Hermitian
9 SPECIAL MATRICES
where X = [X(0), X(1), · · · , x(N − 1)]T . The IDFT is similarly given as
x = W−1
N X.
(407)
Some properties of WN exist:
If WN = e
−j2π
N
W−1
N
=
WN W∗N
W∗N
=
=
1
W∗
N N
NI
WH
N
(408)
(409)
(410)
, then [23]
m+N/2
WN
= −WNm
(411)
Notice, the DFT matrix is a Vandermonde Matrix.
The following important relation between the circulant matrix and the discrete Fourier transform (DFT) exists
TC = W−1
N (I ◦ (WN t))WN ,
(412)
where t = [t0 , t1 , · · · , tn−1 ]T is the first row of TC .
9.3
Hermitian Matrices and skew-Hermitian
A matrix A ∈ Cm×n is called Hermitian if
AH = A
For real valued matrices, Hermitian and symmetric matrices are equivalent.
A is Hermitian
A is Hermitian
⇔
⇔
xH Ax ∈ R,
eig(A) ∈ R
∀x ∈ Cn×1
(413)
(414)
Note that
A = B + iC
where B, C are hermitian, then
B=
9.3.1
A + AH
,
2
C=
A − AH
2i
Skew-Hermitian
A matrix A is called skew-hermitian if
A = −AH
For real valued matrices, skew-Hermitian and skew-symmetric matrices are
equivalent.
A Hermitian
A skew-Hermitian
A skew-Hermitian
⇔ iA is skew-hermitian
⇔ xH Ay = −xH AH y, ∀x, y
⇒ eig(A) = iλ,
λ∈R
(415)
(416)
(417)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 48
9.4
Idempotent Matrices
9.4
9 SPECIAL MATRICES
Idempotent Matrices
A matrix A is idempotent if
AA = A
Idempotent matrices A and B, have the following properties
An
I−A
AH
I − AH
=
A,
forn = 1, 2, 3, ...
is idempotent
(418)
(419)
is idempotent
is idempotent
(420)
(421)
⇒ AB is idempotent
= Tr(A)
(422)
(423)
A(I − A)
=
0
(424)
(I − A)A
A+
=
=
0
A
(425)
(426)
f (sI + tA)
=
(I − A)f (s) + Af (s + t)
(427)
If AB = BA
rank(A)
Note that A − I is not necessarily idempotent.
9.4.1
Nilpotent
A matrix A is nilpotent if
A2 = 0
A nilpotent matrix has the following property:
f (sI + tA)
9.4.2
=
If (s) + tAf ′ (s)
(428)
Unipotent
A matrix A is unipotent if
AA = I
A unipotent matrix has the following property:
f (sI + tA)
9.5
=
[(I + A)f (s + t) + (I − A)f (s − t)]/2
(429)
Orthogonal matrices
If a square matrix Q is orthogonal, if and only if,
QT Q = QQT = I
and then Q has the following properties
• Its eigenvalues are placed on the unit circle.
• Its eigenvectors are unitary, i.e. have length one.
• The inverse of an orthogonal matrix is orthogonal too.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 49
9.6
Positive Definite and Semi-definite Matrices
9 SPECIAL MATRICES
Basic properties for the orthogonal matrix Q
9.5.1
Q−1
Q−T
=
=
QT
Q
QQT
QT Q
=
=
I
I
det(Q)
=
±1
Ortho-Sym
A matrix Q+ which simultaneously is orthogonal and symmetric is called an
ortho-sym matrix [20]. Hereby
QT+ Q+
Q+
=
I
(430)
=
QT+
(431)
The powers of an ortho-sym matrix are given by the following rule
Qk+
=
=
9.5.2
1 + (−1)k
1 + (−1)k+1
I+
Q+
2
2
1 + cos(kπ)
1 − cos(kπ)
I+
Q+
2
2
(432)
(433)
Ortho-Skew
A matrix which simultaneously is orthogonal and antisymmetric is called an
ortho-skew matrix [20]. Hereby
QH
− Q−
Q−
=
I
(434)
=
−QH
−
(435)
The powers of an ortho-skew matrix are given by the following rule
Qk−
=
=
9.5.3
ik − (−i)k
ik + (−i)k
I−i
Q−
2
2
π
π
cos(k )I + sin(k )Q−
2
2
(436)
(437)
Decomposition
A square matrix A can always be written as a sum of a symmetric A+ and an
antisymmetric matrix A−
A = A+ + A−
(438)
9.6
9.6.1
Positive Definite and Semi-definite Matrices
Definitions
A matrix A is positive definite if and only if
xT Ax > 0,
∀x 6= 0
(439)
∀x
(440)
A matrix A is positive semi-definite if and only if
xT Ax ≥ 0,
Note that if A is positive definite, then A is also positive semi-definite.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 50
9.6
Positive Definite and Semi-definite Matrices
9.6.2
9 SPECIAL MATRICES
Eigenvalues
The following holds with respect to the eigenvalues:
A pos. def.
A pos. semi-def.
9.6.3
H
⇔ eig( A+A
)>0
2
H
⇔ eig( A+A
)≥0
2
(441)
Trace
The following holds with respect to the trace:
A pos. def.
A pos. semi-def.
9.6.4
⇒
⇒
Tr(A) > 0
Tr(A) ≥ 0
(442)
Inverse
If A is positive definite, then A is invertible and A−1 is also positive definite.
9.6.5
Diagonal
If A is positive definite, then Aii > 0, ∀i
9.6.6
Decomposition I
The matrix A is positive semi-definite of rank r ⇔ there exists a matrix B of
rank r such that A = BBT
The matrix A is positive definite ⇔ there exists an invertible matrix B such
that A = BBT
9.6.7
Decomposition II
Assume A is an n × n positive semi-definite, then there exists an n × r matrix
B of rank r such that BT AB = I.
9.6.8
Equation with zeros
Assume A is positive semi-definite, then XT AX = 0
9.6.9
⇒
AX = 0
Rank of product
Assume A is positive definite, then rank(BABT ) = rank(B)
9.6.10
Positive definite property
If A is n × n positive definite and B is r × n of rank r, then BABT is positive
definite.
9.6.11
Outer Product
If X is n × r, where n ≤ r and rank(X) = n, then XXT is positive definite.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 51
9.7
Singleentry Matrix, The
9.6.12
9 SPECIAL MATRICES
Small pertubations
If A is positive definite and B is symmetric, then A − tB is positive definite for
sufficiently small t.
9.6.13
Hadamard inequality
If A is a positive definite or semi-definite matrix, then
Y
det(A) ≤
Aii
i
See [15, pp.477]
9.6.14
Hadamard product relation
Assume that P = AAT and Q = BBT are semi positive definite matrices, it
then holds that
P ◦ Q = RRT
where the columns of R are constructed as follows: ri+(j−1)NA = ai ◦ bj , for
i = 1, 2, ..., NA and j = 1, 2, ..., NB . The result is unpublished, but reported by
Pavel Sakov and Craig Bishop.
9.7
9.7.1
Singleentry Matrix, The
Definition
The single-entry matrix Jij ∈ Rn×n is defined as the matrix which is zero
everywhere except in the entry (i, j) in which it is 1. In a 4 × 4 example one
might have
0 0 0 0
0 0 1 0
J23 =
(443)
0 0 0 0
0 0 0 0
The single-entry matrix is very useful when working with derivatives of expressions involving matrices.
9.7.2
Swap and Zeros
Assume A to be n × m and Jij to be m × p
AJij = 0 0 . . . Ai
...
0
(444)
i.e. an n × p matrix of zeros with the i.th column of A in place of the j.th
column. Assume A to be n × m and Jij to be p × n
0
..
.
0
(445)
Jij A =
Aj
0
.
..
0
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 52
9.7
Singleentry Matrix, The
9 SPECIAL MATRICES
i.e. an p × m matrix of zeros with the j.th row of A in the placed of the i.th
row.
9.7.3
Rewriting product of elements
Aki Bjl = (Aei eTj B)kl
=
T
Aik Blj = (A ei eTj BT )kl
Aik Bjl = (AT ei eTj B)kl
Aki Blj =
9.7.4
=
=
(Aei eTj BT )kl
=
(AJij B)kl
T
ij
T
ij
T
(A J B )kl
(A J B)kl
ij
T
(AJ B )kl
(446)
(447)
(448)
(449)
Properties of the Singleentry Matrix
If i = j
Jij Jij = Jij
(Jij )T (Jij )T = Jij
Jij (Jij )T = Jij
If i 6= j
Jij Jij = 0
(Jij )T Jij = Jij
(Jij )T (Jij )T = 0
Jij (Jij )T = Jii
9.7.5
(Jij )T Jij = Jjj
The Singleentry Matrix in Scalar Expressions
Assume A is n × m and J is m × n, then
Tr(AJij ) = Tr(Jij A) = (AT )ij
(450)
Assume A is n × n, J is n × m and B is m × n, then
Tr(AJij B)
=
(AT BT )ij
(451)
Tr(AJji B)
Tr(AJij Jij B)
=
=
(BA)ij
diag(AT BT )ij
(452)
(453)
Assume A is n × n, Jij is n × m B is m × n, then
xT AJij Bx
x AJij Jij Bx
=
=
T
9.7.6
(AT xxT BT )ij
diag(AT xxT BT )ij
(454)
(455)
Structure Matrices
The structure matrix is defined by
∂A
= Sij
∂Aij
(456)
Sij = Jij
(457)
Sij = Jij + Jji − Jij Jij
(458)
If A has no special structure then
If A is symmetric then
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 53
9.8
Symmetric, Skew-symmetric/Antisymmetric
9.8
9.8.1
9 SPECIAL MATRICES
Symmetric, Skew-symmetric/Antisymmetric
Symmetric
The matrix A is said to be symmetric if
A = AT
(459)
Symmetric matrices have many important properties, e.g. that their eigenvalues
are real and eigenvectors orthogonal.
9.8.2
Skew-symmetric/Antisymmetric
The antisymmetric matrix is also known as the skew symmetric matrix. It has
the following property from which it is defined
A = −AT
(460)
Hereby, it can be seen that the antisymmetric matrices always have a zero
diagonal. The n × n antisymmetric matrices also have the following properties.
det(AT )
=
det(−A) = (−1)n det(A)
(461)
− det(A)
=
det(−A) = 0,
(462)
if n is odd
The eigenvalues of an antisymmetric matrix are placed on the imaginary axis
and the eigenvectors are unitary.
9.8.3
Decomposition
A square matrix A can always be written as a sum of a symmetric A+ and an
antisymmetric matrix A−
A = A+ + A−
(463)
Such a decomposition could e.g. be
A=
9.9
A − AT
A + AT
+
= A+ + A−
2
2
(464)
Toeplitz Matrices
A Toeplitz matrix T is a matrix where the elements of each diagonal is the
same. In the n × n square case, it has the following structure:
t11 t12 · · · t1n
t0
t1 · · · tn−1
..
..
..
..
t21 . . . . . .
.
.
.
.
= t−1
T=
(465)
.
..
..
..
..
..
..
.
. t12
.
.
t1
.
tn1 · · · t21 t11
t−(n−1) · · · t−1
t0
A Toeplitz matrix is persymmetric. If a matrix is persymmetric (or orthosymmetric), it means that the matrix is symmetric about its northeast-southwest
diagonal (anti-diagonal) [12]. Persymmetric matrices is a larger class of matrices, since a persymmetric matrix not necessarily has a Toeplitz structure. There
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 54
9.10
Transition matrices
9 SPECIAL MATRICES
are some special cases of Toeplitz matrices. The symmetric Toeplitz matrix is
given by:
t0
t1 · · · tn−1
..
..
..
t1
.
.
.
(466)
T=
.
..
..
..
.
.
t1
tn−1 · · · t1
t0
The circular Toeplitz matrix:
t0
tn−1
TC =
.
..
t1
t1
..
.
..
.
···
The upper triangular Toeplitz matrix:
t 0 t1
0 ...
TU =
. .
..
..
0 ···
···
..
.
..
.
tn−1
···
..
.
..
.
0
and the lower triangular Toeplitz matrix:
t0
0
.
..
t−1
TL =
..
..
.
.
t−(n−1) · · ·
9.9.1
tn−1
..
.
t1
t0
tn−1
..
.
,
t1
t0
···
..
.
..
.
t−1
0
..
.
0
t0
(467)
(468)
(469)
Properties of Toeplitz Matrices
The Toeplitz matrix has some computational advantages. The addition of two
Toeplitz matrices can be done with O(n) flops, multiplication of two Toeplitz
matrices can be done in O(n ln n) flops. Toeplitz equation systems can be solved
in O(n2 ) flops. The inverse of a positive definite Toeplitz matrix can be found
in O(n2 ) flops too. The inverse of a Toeplitz matrix is persymmetric. The
product of two lower triangular Toeplitz matrices is a Toeplitz matrix. More
information on Toeplitz matrices and circulant matrices can be found in [13, 7].
9.10
Transition matrices
A square matrix P is a transition matrix, also known as stochastic matrix or
probability matrix, if
X
(P)ij = 1
0 ≤ (P)ij ≤ 1,
j
The transition matrix usually describes the probability of moving from state i
to j in one step and is closely related to markov processes. Transition matrices
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 55
9.11
Units, Permutation and Shift
9 SPECIAL MATRICES
have the following properties
Prob[i → j in 1 step]
Prob[i → j in 2 steps]
Prob[i → j in k steps]
If all rows are identical
αP
=
=
(P)ij
(P2 )ij
(470)
(471)
=
(Pk )ij
(472)
n
⇒ P =P
= α,
α is called invariant
where α is a so-called stationary probability vector, i.e., 0 ≤ αi ≤ 1 and
1.
9.11
(473)
(474)
P
i
αi =
Units, Permutation and Shift
9.11.1
Unit vector
Let ei ∈ Rn×1 be the ith unit vector, i.e. the vector which is zero in all entries
except the ith at which it is 1.
9.11.2
Rows and Columns
i.th row of A
j.th column of A
9.11.3
=
=
eTi A
Aej
(475)
(476)
Permutations
Let P be some permutation
0 1
P= 1 0
0 0
matrix, e.g.
0
0 = e2
1
e1
e3
For permutation matrices it holds that
eT2
= eT1
eT3
PPT = I
and that
AP =
Ae2
Ae1
Ae3
(478)
eT2 A
PA = eT1 A
eT3 A
(477)
(479)
That is, the first is a matrix which has columns of A but in permuted sequence
and the second is a matrix which has the rows of A but in the permuted sequence.
9.11.4
Translation, Shift or Lag Operators
Let L denote the lag (or ’translation’
example by
0
1
L=
0
0
or ’shift’) operator defined on a 4 × 4
0
0
1
0
0
0
0
1
0
0
0
0
(480)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 56
9.12
Vandermonde Matrices
9 SPECIAL MATRICES
i.e. a matrix of zeros with one on the sub-diagonal, (L)ij = δi,j+1 . With some
signal xt for t = 1, ..., N , the n.th power of the lag operator shifts the indices,
i.e.
n 0
for t = 1, .., n
(Ln x)t =
(481)
xt−n for t = n + 1, ..., N
A related but slightly different matrix is the ’recurrent shifted’ operator defined
on a 4x4 example by
0 0 0 1
1 0 0 0
(482)
L̂ =
0 1 0 0
0 0 1 0
i.e. a matrix defined by (L̂)ij = δi,j+1 + δi,1 δj,dim(L) . On a signal x it has the
effect
(483)
(L̂n x)t = xt′ , t′ = [(t − n) mod N ] + 1
That is, L̂ is like the shift operator L except that it ’wraps’ the signal as if it
was periodic and shifted (substituting the zeros with the rear end of the signal).
Note that L̂ is invertible and orthogonal, i.e.
L̂−1 = L̂T
9.12
(484)
Vandermonde Matrices
A Vandermonde matrix has the form [15]
1 v1 v12 · · ·
1 v2 v22 · · ·
V= . .
..
.. ..
.
1 vn vn2 · · ·
v1n−1
v2n−1
..
.
vnn−1
.
(485)
The transpose of V is also said to a Vandermonde matrix. The determinant is
given by [29]
Y
det V =
(vi − vj )
(486)
i>j
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 57
10
10
10.1
10.1.1
FUNCTIONS AND OPERATORS
Functions and Operators
Functions and Series
Finite Series
(Xn − I)(X − I)−1 = I + X + X2 + ... + Xn−1
10.1.2
(487)
Taylor Expansion of Scalar Function
Consider some scalar function f (x) which takes the vector x as an argument.
This we can Taylor expand around x0
1
f (x) ∼
= f (x0 ) + g(x0 )T (x − x0 ) + (x − x0 )T H(x0 )(x − x0 )
2
where
g(x0 ) =
10.1.3
∂f (x)
∂x
H(x0 ) =
x0
∂ 2 f (x)
∂x∂xT
(488)
x0
Matrix Functions by Infinite Series
As for analytical functions in one dimension, one can define a matrix function
for square matrices X by an infinite series
f (X) =
∞
X
c n Xn
(489)
n=0
P
assuming the limit exists and is finite. If the coefficients cn fulfils n cn xn < ∞,
then one can prove that the above series exists and is finite, see [1]. Thus for
any analytical function f (x) there exists a corresponding matrix function f (x)
constructed by the Taylor expansion. Using this one can prove the following
results:
1) A matrix A is a zero of its own characteristic polynomium [1]:
X
p(λ) = det(Iλ − A) =
c n λn
⇒
p(A) = 0
(490)
n
2) If A is square it holds that [1]
A = UBU−1
⇒
f (A) = Uf (B)U−1
(491)
3) A useful fact when using power series is that
An → 0forn → ∞
10.1.4
if
|A| < 1
(492)
Identity and commutations
It holds for an analytical matrix function f (X) that
f (AB)A = Af (BA)
(493)
see B.1.2 for a proof.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 58
10.2
Kronecker and Vec Operator
10.1.5
10
FUNCTIONS AND OPERATORS
Exponential Matrix Function
In analogy to the ordinary scalar exponential function, one can define exponential and logarithmic matrix functions:
eA
≡
e−A
≡
etA
≡
ln(I + A)
≡
∞
X
1 n
1
A = I + A + A2 + ...
n!
2
n=0
∞
X
1
1
(−1)n An = I − A + A2 − ...
n!
2
n=0
∞
X
1
1
(tA)n = I + tA + t2 A2 + ...
n!
2
n=0
∞
X
1
1
(−1)n−1 n
A = A − A2 + A3 − ...
n
2
3
n=1
(494)
(495)
(496)
(497)
Some of the properties of the exponential function are [1]
eA eB
(eA )−1
d tA
e
dt
d
Tr(etA )
dt
det(eA )
10.1.6
10.2
10.2.1
=
=
eA+B
e−A
=
AetA = etA A,
=
Tr(AetA )
(501)
=
eTr(A)
(502)
if
AB = BA
t∈R
(498)
(499)
(500)
Trigonometric Functions
sin(A)
≡
cos(A)
≡
∞
X
(−1)n A2n+1
1
1
= A − A3 + A5 − ...
(2n
+
1)!
3!
5!
n=0
∞
X
1
1
(−1)n A2n
= I − A2 + A4 − ...
(2n)!
2!
4!
n=0
(503)
(504)
Kronecker and Vec Operator
The Kronecker Product
The Kronecker product of an m × n matrix A and an r × q matrix B, is an
mr × nq matrix, A ⊗ B defined as
A11 B A12 B ... A1n B
A21 B A22 B ... A2n B
(505)
A⊗B=
..
..
.
.
Am1 B
Am2 B
...
Amn B
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 59
10.2
Kronecker and Vec Operator
10
FUNCTIONS AND OPERATORS
The Kronecker product has the following properties (see [19])
A ⊗ (B + C)
A⊗B
=
6=
(A ⊗ B)T
(A ⊗ B)(C ⊗ D)
=
=
A ⊗ (B ⊗ C)
(αA A ⊗ αB B)
=
=
A⊗B+A⊗C
B⊗A
in general
(506)
(507)
(A ⊗ B) ⊗ C
αA αB (A ⊗ B)
(508)
(509)
A−1 ⊗ B−1
(512)
A T ⊗ BT
AC ⊗ BD
(510)
(511)
(A ⊗ B)−1
=
+
(A ⊗ B)
rank(A ⊗ B)
=
=
A ⊗B
rank(A)rank(B)
(513)
(514)
Tr(A ⊗ B)
=
(515)
det(A ⊗ B)
{eig(A ⊗ B)}
=
=
Tr(A)Tr(B) = Tr(ΛA ⊗ ΛB )
det(A)rank(B) det(B)rank(A)
{eig(A ⊗ B)}
=
eig(A ⊗ B)
=
+
+
{eig(B ⊗ A)}
if A, B are square
{eig(A)eig(B)T }
if A, B are symmetric and square
eig(A) ⊗ eig(B)
(516)
(517)
(518)
(519)
Where {λi } denotes the set of values λi , that is, the values in no particular
order or structure, and ΛA denotes the diagonal matrix with the eigenvalues of
A.
10.2.2
The Vec Operator
The vec-operator applied on a matrix A stacks the columns into a vector, i.e.
for a 2 × 2 matrix
A11
A21
A11 A12
A=
vec(A) =
A12
A21 A22
A22
Properties of the vec-operator include (see [19])
vec(AXB)
Tr(AT B)
=
=
(BT ⊗ A)vec(X)
vec(A)T vec(B)
(520)
(521)
vec(A + B)
=
vec(A) + vec(B)
(522)
vec(αA)
aT XBXT c
=
=
α · vec(A)
vec(X)T (B ⊗ caT )vec(X)
(523)
(524)
See B.1.1 for a proof for Eq. 524.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 60
10.3
Vector Norms
10.3
10.3.1
10
FUNCTIONS AND OPERATORS
Vector Norms
Examples
||x||1
=
||x||22
=
||x||p
=
||x||∞
=
X
i
H
|xi |
x x
"
X
i
(525)
(526)
|xi |p
#1/p
(527)
max |xi |
(528)
i
Further reading in e.g. [12, p. 52]
10.4
10.4.1
Matrix Norms
Definitions
A matrix norm is a mapping which fulfils
||A||
||A||
≥
=
||A + B||
≤
||cA||
10.4.2
=
0
0⇔A=0
|c|||A||,
||A|| + ||B||
(529)
(530)
c∈R
(531)
(532)
Induced Norm or Operator Norm
An induced norm is a matrix norm induced by a vector norm by the following
||A|| = sup{||Ax||
|
||x|| = 1}
(533)
where || · || on the left side is the induced matrix norm, while || · || on the right
side denotes the vector norm. For induced norms it holds that
||I||
||Ax||
||AB||
10.4.3
=
≤
≤
1
||A|| · ||x||,
||A|| · ||B||,
for all A, x
for all A, B
(534)
(535)
(536)
Examples
||A||1
=
||A||2
||A||p
=
=
||A||∞
=
||A||F
=
X
max
|Aij |
j
i
q
max eig(AH A)
( max ||Ax||p )1/p
||x||p =1
X
|Aij |
max
i
j
sX
q
|Aij |2 = Tr(AAH )
(537)
(538)
(539)
(540)
(Frobenius)
(541)
ij
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 61
10.5
Rank
10
||A||max
||A||KF
=
=
max |Aij |
FUNCTIONS AND OPERATORS
(542)
ij
||sing(A)||1
(Ky Fan)
(543)
where sing(A) is the vector of singular values of the matrix A.
10.4.4
Inequalities
E. H. Rasmussen has in yet unpublished material derived and collected the
following inequalities. They are collected in a table as below, assuming A is an
m × n, and d = rank(A)
||A||max
||A||1
||A||∞
||A||2
||A||F
||A||KF
||A||max
m
√n
mn
√
√ mn
mnd
||A||1
1
||A||∞
1
m
√n
n
√
√n
nd
√
m
√
√m
md
||A||2
√1
√m
n
√
d
d
||A||F
√1
√m
n
1
√
||A||KF
√1
√m
n
1
1
d
which are to be read as, e.g.
||A||2 ≤
√
m · ||A||∞
(544)
10.4.5
Condition Number
p
The 2-norm of A equals (max(eig(AT A))) [12, p.57]. For a symmetric, positive definite matrix, this reduces to max(eig(A)) The condition number based
on the 2-norm thus reduces to
kAk2 kA−1 k2 = max(eig(A)) max(eig(A−1 )) =
10.5
10.5.1
max(eig(A))
.
min(eig(A))
(545)
Rank
Sylvester’s Inequality
If A is m × n and B is n × r, then
rank(A) + rank(B) − n ≤ rank(AB) ≤ min{rank(A), rank(B)}
10.6
(546)
Integral Involving Dirac Delta Functions
Assuming A to be square, then
Z
p(s)δ(x − As)ds =
1
p(A−1 x)
det(A)
Assuming A to be ”underdetermined”, i.e. ”tall”, then
)
(
Z
√ 1 T p(A+ x) if x = AA+ x
det(A A)
p(s)δ(x − As)ds =
0
elsewhere
(547)
(548)
See [9].
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 62
10.7
Miscellaneous
10.7
10
FUNCTIONS AND OPERATORS
Miscellaneous
For any A it holds that
rank(A) = rank(AT ) = rank(AAT ) = rank(AT A)
(549)
It holds that
A is positive definite
⇔
∃B invertible, such that A = BBT
(550)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 63
A ONE-DIMENSIONAL RESULTS
A
A.1
A.1.1
One-dimensional Results
Gaussian
Density
p(x) = √
A.1.2
A.1.3
Normalization
Z
(s−µ)2
e− 2σ2 ds
Z
2
e−(ax +bx+c) dx
Z
2
ec2 x +c1 x+c0 dx
=
=
=
(551)
√
2πσ 2
r
2
b − 4ac
π
exp
a
4a
2
r
c1 − 4c2 c0
π
exp
−c2
−4c2
(552)
(553)
(554)
Derivatives
∂p(x)
∂µ
∂ ln p(x)
∂µ
∂p(x)
∂σ
∂ ln p(x)
∂σ
A.1.4
(x − µ)2
exp −
2σ 2
2πσ 2
1
=
=
=
=
(x − µ)
σ2
(x − µ)
σ2
1 (x − µ)2
p(x)
−
1
σ
σ2
1 (x − µ)2
−
1
σ
σ2
p(x)
(555)
(556)
(557)
(558)
Completing the Squares
c2 x2 + c1 x + c0 = −a(x − b)2 + w
−a = c2
b=
1 c1
2 c2
or
c2 x2 + c 1 x + c 0 = −
µ=
A.1.5
−c1
2c2
σ2 =
w=
1 c21
+ c0
4 c2
1
(x − µ)2 + d
2σ 2
−1
2c2
d = c0 −
c21
4c2
Moments
If the density is expressed by
(s − µ)2
1
exp −
p(x) = √
2σ 2
2πσ 2
or
p(x) = C exp(c2 x2 + c1 x)
(559)
then the first few basic moments are
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 64
A.2
One Dimensional Mixture of Gaussians
A ONE-DIMENSIONAL RESULTS
hxi
=
µ
=
=
σ 2 + µ2
=
hx3 i
=
3σ 2 µ + µ3
=
hx4 i
=
µ4 + 6µ2 σ 2 + 3σ 4
=
hx2 i
−c1
2c2
2
−c1
−1
2c2 + h 2c2
i
c21
c1
3
−
2
(2c )
2c2
2
2 4
c1
c1
+
6
2c2
2c2
and the central moments are
h(x − µ)i
=
2
h(x − µ) i
h(x − µ)3 i
h(x − µ)4 i
0
=
2
=
=
σ
0
=
3σ 4
=
=
h0
=
3
−1
2c2
0
h
i
1
2c2
−1
2c2
+3
1
2c2
2
i2
A kind of pseudo-moments (un-normalized integrals) can easily be derived as
2
r
Z
π
c1
exp
hxn i
(560)
exp(c2 x2 + c1 x)xn dx = Zhxn i =
−c2
−4c2
¿From the un-centralized moments one can derive other entities like
hx2 i − hxi2
hx3 i − hx2 ihxi
hx4 i − hx2 i2
A.2
A.2.1
=
=
σ2
2σ 2 µ
=
=
=
2σ 4 + 4µ2 σ 2
=
h
c2
1 − 4 2c12
i
One Dimensional Mixture of Gaussians
Density and Normalization
p(s) =
K
X
k
A.2.2
−1
2c2
2c1
(2c2 )2
2
(2c2 )2
Moments
p
ρk
1 (s − µk )2
exp −
2
σk2
2πσk2
(561)
A useful fact of MoG, is that
hxn i =
X
k
ρk hxn ik
(562)
where h·ik denotes average with respect to the k.th component. We can calculate
the first four moments from the densities
X
1 (x − µk )2
1
(563)
ρk p
p(x) =
exp −
2
σk2
2πσk2
k
X
ρk Ck exp ck2 x2 + ck1 x
(564)
p(x) =
k
as
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 65
B
hxi
2
hx i
=
=
hx3 i
=
hx4 i
=
P
k
ρk µ k
=
P
2
k ρk (σk
2
3
k ρk (3σk µk + µk )
=
P
4
2 2
4
k ρk (µk + 6µk σk + 3σk )
=
P
+
µ2k )
=
P
P
P
P
k
k
ρk
ρk
k ρk
k ρk
PROOFS AND DETAILS
h
−ck1
2ck2
h
−1
2ck2
i
+
ck1
2
(2ck2 )
1
2ck2
If all the gaussians are centered, i.e. µk = 0 for all k, then
hxi
=
hx2 i
hx3 i
=
=
hx4 i
=
0
P
0
P
=
2
k ρk σ k
k
ρk 3σk4
=
=
=
0
P
0
P
h
k
ρk
k
ρk 3
−1
2ck2
h
i
−1
2ck2
h
−ck1
2ck2
3−
2
2
c2k1
2ck2
ck1
2ck2
ii
2
c2
− 6 2ck1
+3
k2
i2
¿From the un-centralized moments one can derive other entities like
2
P
2
′
′
hx2 i − hxi2
=
k,k′ ρk ρk µk + σk − µk µk
P
2
2
2
3
hx3 i − hx2 ihxi =
′ ρk ρk ′ 3σk µk + µk − (σk + µk )µk ′
k,k
4
P
2
2
4
2
2
2 2
hx4 i − hx2 i2
=
k,k′ ρk ρk′ µk + 6µk σk + 3σk − (σk + µk )(σk′ + µk′ )
A.2.3
Derivatives
P
Defining p(s) = k ρk Ns (µk , σk2 ) we get for a parameter θj of the j.th component
ρj Ns (µj , σj2 ) ∂ ln(ρj Ns (µj , σj2 ))
∂ ln p(s)
=P
(565)
2
∂θj
∂θj
k ρk Ns (µk , σk )
that is,
∂ ln p(s)
∂ρj
=
∂ ln p(s)
∂µj
=
∂ ln p(s)
∂σj
=
ρj Ns (µj , σj2 ) 1
P
2
k ρk Ns (µk , σk ) ρj
ρj Ns (µj , σj2 ) (s − µj )
P
2
σj2
k ρk Ns (µk , σk )
#
"
ρj Ns (µj , σj2 ) 1 (s − µj )2
P
−1
2
σj2
k ρk Ns (µk , σk ) σj
(566)
(567)
(568)
Note thatP
ρk must be constrained to be proper ratios. Defining the ratios by
ρj = erj / k erk , we obtain
∂ρl
∂ ln p(s) X ∂ ln p(s) ∂ρl
=
where
= ρl (δlj − ρj )
(569)
∂rj
∂ρl ∂rj
∂rj
l
B
B.1
B.1.1
Proofs and Details
Misc Proofs
Proof of Equation 524
The following proof is work of Florian Roemer. Note the the vectors and matrices below can be complex and the notation XH is used for transpose and
conjugated, while XT is only transpose of the complex matrix.
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 66
B.1 Misc Proofs
B
PROOFS AND DETAILS
Define the row vector y = aH XB and the column vector z = XH c. Then
aT XBXT c = yz = zT yT
Note that y can be rewritten as vec(y)T which is the same as
vec(conj(y))H = vec(aT conj(X)conj(B))H
where ”conj” means complex conjugated. Applying the vec rule for linear forms
Eq 520, we get
y = (BH ⊗ aT vec(conj(X))H = vec(X)T (B ⊗ conj(a))
where we have also used the rule for transpose of Kronecker products. For yT
this yields (BT ⊗ aH )vec(X). Similarly we can rewrite z which is the same as
vec(zT ) = vec(cT conj(X)). Applying again Eq 520, we get
z = (I ⊗ cT )vec(conj(X))
where I is the identity matrix. For zT we obtain vec(X)(I ⊗ c). Finally, the
original expression is zT yT which now takes the form
vec(X)H (I ⊗ c)(BT ⊗ aH )vec(X)
the final step is to apply the rule for products of Kronecker products and by
that combine the Kronecker products. This gives
vec(X)H (BT ⊗ caH )vec(X)
which is the desired result.
B.1.2
Proof of Equation 493
For any analytical function f (X) of a matrix argument X, it holds that
!
∞
X
n
cn (AB)
A
f (AB)A =
n=0
=
=
∞
X
n=0
∞
X
cn (AB)n A
cn A(BA)n
n=0
∞
X
cn (BA)n
=
A
=
Af (BA)
n=0
B.1.3
Proof of Equation 91
Essentially we need to calculate
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 67
B.1 Misc Proofs
B
∂(Xn )kl
∂Xij
∂
∂Xij
=
=
X
PROOFS AND DETAILS
Xk,u1 Xu1 ,u2 ...Xun−1 ,l
u1 ,...,un−1
δk,i δu1 ,j Xu1 ,u2 ...Xun−1 ,l
+Xk,u1 δu1 ,i δu2 ,j ...Xun−1 ,l
..
.
+Xk,u1 Xu1 ,u2 ...δun−1 ,i δl,j
=
n−1
X
(Xr )ki (Xn−1−r )jl
r=0
=
n−1
X
(Xr Jij Xn−1−r )kl
r=0
Using the properties of the single entry matrix found in Sec. 9.7.4, the result
follows easily.
B.1.4
Details on Eq. 571
∂ det(XH AX)
=
=
det(XH AX)Tr[(XH AX)−1 ∂(XH AX)]
det(XH AX)Tr[(XH AX)−1 (∂(XH )AX + XH ∂(AX))]
=
det(XH AX) Tr[(XH AX)−1 ∂(XH )AX]
+Tr[(XH AX)−1 XH ∂(AX)]
=
det(XH AX) Tr[AX(XH AX)−1 ∂(XH )]
+Tr[(XH AX)−1 XH A∂(X)]
First, the derivative is found with respect to the real part of X
Tr[AX(XH AX)−1 ∂(XH )]
∂ det(XH AX)
= det(XH AX)
∂ℜX
∂ℜX
Tr[(XH AX)−1 XH A∂(X)]
+
∂ℜX
= det(XH AX) AX(XH AX)−1 + ((XH AX)−1 XH A)T
Through the calculations, (100) and (240) were used. In addition, by use of
(241), the derivative is found with respect to the imaginary part of X
Tr[AX(XH AX)−1 ∂(XH )]
∂ det(XH AX)
i
= i det(XH AX)
∂ℑX
∂ℑX
Tr[(XH AX)−1 XH A∂(X)]
+
∂ℑX
H
= det(X AX) AX(XH AX)−1 − ((XH AX)−1 XH A)T
Hence, derivative yields
∂ det(XH AX)
∂X
=
=
1 ∂ det(XH AX)
∂ det(XH AX)
−i
2
∂ℜX
∂ℑX
T
H
H
−1 H
det(X AX) (X AX) X A
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 68
B.1 Misc Proofs
B
PROOFS AND DETAILS
and the complex conjugate derivative yields
∂ det(XH AX)
∂X∗
=
=
1 ∂ det(XH AX)
∂ det(XH AX)
+i
2
∂ℜX
∂ℑX
H
H
−1
det(X AX)AX(X AX)
Notice, for real X, A, the sum of (249) and (250) is reduced to (54).
Similar calculations yield
∂ det(XAXH )
∂X
=
=
and
∂ det(XAXH )
∂X∗
=
=
1 ∂ det(XAXH )
∂ det(XAXH )
−i
2
∂ℜX
∂ℑX
H
H
H −1 T
det(XAX ) AX (XAX )
1 ∂ det(XAXH )
∂ det(XAXH )
+i
2
∂ℜX
∂ℑX
det(XAXH )(XAXH )−1 XA
(570)
(571)
Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 69
REFERENCES
REFERENCES
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Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 71
Index
Anti-symmetric, 54
Normal-Inverse Gamma distribution, 37
Normal-Inverse Wishart distribution, 39
Block matrix, 46
Orthogonal, 49
Chain rule, 15
Cholesky-decomposition, 32
Co-kurtosis, 34
Co-skewness, 34
Condition number, 62
Cramers Rule, 29
Power series of matrices, 58
Probability matrix, 55
Pseudo-inverse, 21
Schur complement, 41, 47
Single entry matrix, 52
Singular Valued Decomposition (SVD),
31
Skew-Hermitian, 48
Skew-symmetric, 54
Stochastic matrix, 55
Student-t, 37
Sylvester’s Inequality, 62
Symmetric, 54
Derivative of a complex matrix, 24
Derivative of a determinant, 8
Derivative of a trace, 12
Derivative of an inverse, 9
Derivative of symmetric matrix, 15
Derivatives of Toeplitz matrix, 16
Dirichlet distribution, 37
Eigenvalues, 30
Eigenvectors, 30
Exponential Matrix Function, 59
Taylor expansion, 58
Toeplitz matrix, 54
Transition matrix, 55
Trigonometric functions, 59
Gaussian, conditional, 40
Gaussian, entropy, 44
Gaussian, linear combination, 41
Gaussian, marginal, 40
Gaussian, product of densities, 42
Generalized inverse, 21
Unipotent, 49
Vandermonde matrix, 57
Vec operator, 59, 60
Wishart distribution, 38
Woodbury identity, 18
Hadamard inequality, 52
Hermitian, 48
Idempotent, 49
Kronecker product, 59
LDL decomposition, 33
LDM-decomposition, 33
Linear regression, 28
LU decomposition, 32
Lyapunov Equation, 30
Moore-Penrose inverse, 21
Multinomial distribution, 37
Nilpotent, 49
Norm of a matrix, 61
Norm of a vector, 61
72
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