The classical singular value decomposition for a matrix A ∈ Cm×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA∗ and A∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices AA and AA. More generally, we consider the matrix triple (A,G, Ĝ), where G ∈ Cm×m, Ĝ ∈ Cn×n ar...

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a...

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a...

A general mathematical formulation of the n × n proper Orthogonal matrix, that corresponds to a rigid rotation in n-dimensional real Euclidean space, is given here. It is shown that a rigid rotation depends on an angle (principal angle) and on a set of (n−2) principal axes. The latter, however, can be more conveniently replaced by only 2 Orthogonal directions that identify the plane of rotation...

where τ0 = 1. The τ -functions τn are given by the pfaffians of certain skew-symmetric matrix. We identify one-soliton solution as an element of the Weyl group of D-type, and discuss a general structure of the interaction patterns among the solitons. Soliton solutions are characterized by 4N × 4N skew-symmetric constant matrix which we call the B-matrices. We then find that one can have M -soli...

In this note, we study the notion of structured pseudospectra. We prove that for Toeplitz, circulant and symmetric structures, the structured pseudospectrum equals the unstructured pseudospectrum. We show that this is false for Hermitian and skew-Hermitian structures. We generalize the result to pseudospectra of matrix polynomials. Indeed, we prove that the structured pseudospectrum equals the ...