Eigenvectors of Tensors—A Primer

Abstract

We give an introduction to the theory and to some applications of eigenvectors of tensors (in other words, invariant one-dimensional subspaces of homogeneous polynomial maps), including a review of some concepts that are useful for their discussion. The intent is to give practitioners an overview of fundamental notions, results and techniques.

Appendix

1.1 7.1 Some Linear Algebra

For the sake of completeness we prove the following commonly known fact: For real $n\times n$ matrices $A$ and $B$ one has

$$ \det \begin{pmatrix} A&-B \\ B&A \end{pmatrix} \geq 0. $$

First assume that $A$ is invertible. Then

$$ \begin{pmatrix} A&-B \\ B&A \end{pmatrix} = \begin{pmatrix} A & 0 \\ 0&A \end{pmatrix} \cdot \begin{pmatrix} I_{n} &0 \\ C & I_{n} \end{pmatrix} \cdot \begin{pmatrix} I_{n} &- C \\ 0 & I_{n}+C^{2} \end{pmatrix} $$

where $I_{n}$ denotes the $n\times n$ identity matrix and $C=A^{-1}B$. The determinant of the first factor is equal to $(\det A)^{2}$, hence $>0$; the second factor has determinant one, and for the third one we get

$$ \det \bigl(I_{n}+C^{2}\bigr)=\det (I_{n}+ iC) \cdot \overline{\det (I_{n}+iC)}>0. $$

In the case of non-invertible $A$ apply the argument to $\rho I_{n}+A$ for small $\rho >0$ and use continuity as $\rho \to 0$.

To apply this to real representations of complex linear maps, let $w=x+iy\in \mathbb{C}^{n}$ with $x,\,y\in \mathbb{R}^{n}$, and write a linear map $L$ from $\mathbb{C}^{n}$ to $\mathbb{C}^{n}$ in the form

$$ L= A+iB; \quad L(x+iy)= (Ax-By ) + i (Bx+Ay ); $$

hence $L$ has a real matrix representation

$$ \begin{pmatrix} A&-B \\ B&A \end{pmatrix} . $$

1.2 7.2 A Hybrid Proof of Bezout’s Theorem

This proof uses Proposition 1 as an algebraic tool, and properties of the Brouwer degree on the analytic side. It is not central to the topic of the present paper but its inclusion is unproblematic and it may be seen as informative anyway.

By Proposition 1, in the complex affine space of structure coefficients, the coefficient sets which correspond to homogeneous polynomial maps $Q$ with a nilpotent ($v\neq0$ and $Q(v)=0$) form the hypersurface defined by the resultant. This hypersurface has real codimension two, hence any two points in its complement can be connected by a continuous curve. These points correspond to two homogeneous polynomial maps without nilpotents. By homotopy invariance they have the same Brouwer degree. To summarize, any two homogeneous polynomial maps of degree $m$ without nilpotents have the same degree. For the special map

$$ \widetilde{Q}:\,\mathbb{C}^{n}\to \mathbb{C}^{n},\qquad x \mapsto \begin{pmatrix} x_{1}^{m} \\ \vdots \\ x_{n}^{m} \end{pmatrix} $$

one easily verifies that the number of solutions of $\widetilde{Q}(x)=(1, \ldots ,1)^{\operatorname{tr}}$ is equal to $m^{n}$, with each solution having multiplicity one.