SIAM Journal on Matrix Analysis and Applications

We describe the spectra of certain tridiagonal matrices arising from differential equations commonly used for modeling flocking behavior. In particular we consider systems resulting from allowing an arbitrary boundary condition for the end of a one-...

We generalize the recent invariant polytope algorithm for computing the joint spectral radius and extend it to a wider class of matrix sets. This, in particular, makes the algorithm applicable to sets of matrices that have finitely many spectrum ...

Given a Hermitian matrix pencil $L(z) = zA - B$ with only real eigenvalues that are either of positive or negative type, the distance to a nearest Hermitian pencil outside the class is considered with respect to a specified norm. These problems are ...

We perform a backward error analysis of polynomial eigenvalue problems solved via linearization. Through the use of dual minimal bases, we unify the construction of strong linearizations for many different polynomial bases. By inspecting the prototypical ...

The scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix $A$. We introduce an efficient variant that uses a much smaller squaring factor when $\|A\| \gg 1$ and a subdiagonal Padé approximant of low ...

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $V$ such that $A = V V ^T$. In this paper, we study the CP-matrix approximation problem: for a given symmetric matrix $C$, find a CP matrix $X$, such that $X$ ...

The second-order Arnoldi (SOAR) procedure is an algorithm for computing an orthonormal basis of the second-order Krylov subspace. It has found applications in solving quadratic eigenvalue problems and model order reduction of second-order dynamical systems ...

We consider the inverse eigenvalue problem of reconstructing a doubly stochastic matrix from the given spectrum data. We reformulate this inverse problem as a constrained nonlinear least squares problem over several matrix manifolds, which minimizes the ...

This paper describes a multilevel preconditioning technique for solving sparse symmetric linear systems of equations. This “Multilevel Schur Low-Rank” (MSLR) preconditioner first builds a tree structure $\mathcal{T}$ based on a hierarchical decomposition ...

Given a special kind of symmetric matrix, we present a decomposition technique that preserves its spectrum. We transform this result into an algorithm for graphs. The algorithm disconnects the graph, resulting in smaller matrices, reducing the complexity of ...

We introduce the concept of mode-$k$ generalized eigenvalues and eigenvectors of a tensor and prove some properties of such eigenpairs. In particular, we derive an upper bound for the number of equivalence classes of generalized tensor eigenpairs using ...

We give sufficient conditions on a symmetric tensor $\mathcal{S}\in\mathrm{S}^d\mathbb{F}^n$ to satisfy the following equality: the symmetric rank of $\mathcal{S}$, denoted as $\mathrm{srank\;}\mathcal{S}$, is equal to the rank of $\mathcal{S}$, denoted as $...

Existing hierarchically semi-separable construction algorithms for dense $n \times n$ matrices require as much as $O(n^2)$ peak workspace memory, at a cost of $O(n^2)$ flops. An algorithm is presented which requires $O(n^{1.5})$ peak worskpace memory in ...

In this paper we study multivariate polynomial functions in complex variables and their corresponding symmetric tensor representations. The focus is to find conditions under which such complex polynomials always take real values. We introduce the notion of ...

We deal with the generalized eigenvalue problem $A{\bm x} = \lambda B {\bm x}$ for nonsquare matrix pencils $ A - \lambda B $, where $ A , B \in \mathbb{C}^{m \times n}$ and $m > n$. A major difficulty inherent in this problem is that perturbation to inputs may ...

In an attempt to progress toward proving the conjecture that the numerical range $W(A)$ is a two-spectral set for the matrix $A$, we propose a study of various constants. We review some partial results; many problems are still open. We describe our ...

We introduce new broadcast and receive communicability indices that can be used as global measures of how effectively information is spread in a directed network. Furthermore, we describe fast and effective criteria for the selection of edges to be added to (...

We consider the solution of a system of linear algebraic equations which is obtained from a Raviart--Thomas mixed finite element formulation of Darcy's equations. In [C. E. Powell and D. Silvester, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 718--738], ...