A formula for polynomials with Hermitian matrix argument

We construct and study orthogonal bases of generalized polynomials on the space of Hermitian matrices. They are obtained by the Gram-Schmidt orthogonalization process from the Schur polynomials. A Berezin-Karpelevich type formula is given for these multivariate polynomials. The normalization of the orthogonal polynomials of Hermitian matrix argument and expansions in such polynomials are investigated.     

Backward errors for eigenvalues and eigenvectors of Hermitian,skew-Hermitian,H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.     

Hermitian quadratic matrix polynomials: Solvents and inverse problems

A monic quadratic Hermitian matrix polynomial $L(\lambda )$ can be factorized into a product of two linear matrix polynomials, say $L(\lambda )=(I\lambda -S)(I\lambda -A)$. For the inverse problem of finding a quadratic matrix polynomial with prescribed spectral data (eigenvalues and eigenvectors) it is natural to prescribe a right solvent A and then determine compatible left solvents S. This problem is explored in the present paper. The splitting of the spectrum between real eigenvalues and nonreal conjugate pairs plays an important role. Special attention is paid to the case of real-symmetric quadratic polynomials and the allocation of the canonical sign characteristics as well as the eigenvalues themselves.     

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler     

A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues

This paper describes a new computational procedure for calculating eigenvalues and eigenvectors of a square matrix. The method is based on a matrix function, the sign of a matrix. Eigenvalues and eigenvectors of matrices with distinct eigenvalues and nondefective matrices with repeated roots can be determined in a straightforward manner. Defective matrices require additional calculations.     

Factorization of selfadjoint quadratic matrix polynomials with real spectrum

Factorization theorems, and properties of sets of eigenvectors, are established for regular selfadjoint quatratic matrix polynomials L(λ) whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type. The two linear factors obtained have spectra which are just the eigenvalues of L(λ) of positive and negative types, respectively.     

Hessenberg matrix for sums of Hermitian positive definite matrices and weighted shifts

In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation m-sum and we obtain an expression for it that involves the Cholesky factorization of the corresponding Hermitian positive definite matrices associated with the Hessenberg components.This work extends a method to obtain the Hessenberg matrix of the sum of measures from the Hessenberg matrices of the individual measures, introduced recently by the authors for subnormal matrices, to matrices which are not necessarily subnormal.Moreover, we give some examples and we obtain the explicit formula for the m-sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the m-sum of two not subnormal Hessenberg matrices.     

Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

We develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space L_m(C^n×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in L_m(C^n×n). We present a family of natural norms on the space L_m(C^n×n) and show that the norms on the spaces C^m+1 and C^n×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space L_m(C^n×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space L_m(C^n×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials.     

On the real parts of eigenvalues of a matrix

We apply a theorem of K_y Fan on eigenvalue location to give a new proof of a result of K. J. Palmer concerning the real parts of the eigenvalues of a matrix.     

Bounds for the real eigenvalues of a cascade matrix

Bounds are derived for the real eigenvalues of a special matrix. Matrices of this form arise in the design of two-up one-down cascades for isotope separation.     

Diagonal elements and eigenvalues of a real symmetric matrix

A simple and constructive proof is given for the existence of a real symmetric matrix with prescribed diagonal elements and eigenvalues. Numerically implementable algorithms for constructing such a matrix are discussed.     

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n   partial information on the minimal degree component of the vanishing ideal of the variety of n×n$n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.     

Expansion formulas for the inertias of Hermitian matrix polynomials and matrix pencils of orthogonal projectors

This paper gives a group of expansion formulas for the inertias of Hermitian matrix polynomials A−A², I−A² and A−A³ through some congruence transformations for block matrices, where A is a Hermitian matrix. Then, the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and Hermitian unitary matrices. As applications, the paper establishes necessary and sufficient conditions for many matrix equalities to hold, as well as many inequalities in the Löwner partial ordering to hold.     

A note on simple eigenvalues of matrix polynomials in queueing models with Erlang distributions

In this note, we study anE _k /E _m /1 queueing model where interarrival and service times are Erlang distributions with parametersk andm respectively. We prove that the roots of the characteristic polynomial associated with Erlang distributions are simple if the arrival and service rates are real. Based on this result, a general solution space of vectors for stationary probabilities is easily constructed.     

A sufficient condition that a matrix have eigenvalues with non-zero real parts

If A is a complex matrix let à be the real matrix obtained by replacing the diagonal elements of A by the moduli of their real parts and by replacing the off-diagonal elements by the negative of their moduli. Then we show that if à is an M-matrix the eigenvalues of A have nonzero real parts and, moreover, the moduli of the real parts are bounded below by the minimum of the real parts of the eigenvalues of Ã.