Hermitian Matrix Polynomials Research Articles

Convergence analysis of vector extended locally optimal block preconditioned extended conjugate gradient method for computing extreme eigenvalues

AbstractThis article is concerned with the convergence analysis of an extension of the locally optimal preconditioned conjugate gradient method for the extreme eigenvalue of a Hermitian matrix polynomial that possesses some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J Numer Anal. 2008;46(5):2567–92.). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given to illustrate the convergence behavior.

Positive semidefinite univariate matrix polynomials

We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of size $$n\times n$$ can be written as a sum of squares $$M=Q^TQ$$ , where Q has size $$(n+1)\times n$$ , which was recently proved by Blekherman–Plaumann–Sinn–Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representations $$M=Q^TQ$$ are generically in one-to-one correspondence with the representations of the nonnegative univariate polynomial $$\det (M)$$ as sums of two squares. In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomial M that is positive semidefinite along the real line, is a square, which is known as the matrix Fejer–Riesz Theorem.

A complete characterization of determinantal quadratic polynomials

The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In this paper we provide a necessary and sufficient condition for the existence of monic Hermitian determinantal representation as well as monic symmetric determinantal representation of size 2 for a given quadratic polynomial. Further we propose a method to construct such a monic determinantal representation (MDR) of size 2 if it exists. It is known that a quadratic polynomial f(x)=xTAx+bTx+1 has a symmetric MDR of size n+1 if A is negative semidefinite. We prove that if a quadratic polynomial f(x) with A which is not negative semidefinite has an MDR of size greater than 2, then it has an MDR of size 2 too. Finally, we characterize all quadratic polynomials that exhibit MDRs of any size.

On the sign characteristic of Hermitian linearizations in [formula omitted

The computation of eigenvalues and eigenvectors of matrix polynomials is an important, but difficult, problem. The standard approach to solve this problem is to use linearizations, which are matrix polynomials of degree 1 that share the eigenvalues of P(λ).Hermitian matrix polynomials and their real eigenvalues are of particular interest in applications. Attached to these eigenvalues is a set of signs called the sign characteristic. From both a theoretical and a practical point of view, it is important to be able to recover the sign characteristic of a Hermitian linearization of P(λ) from the sign characteristic of P(λ).In this paper, for a Hermitian matrix polynomial P(λ) with nonsingular leading coefficient, we describe, in terms of the sign characteristic of P(λ), the sign characteristic of the Hermitian linearizations in the vector space DL(P) (Mackey, Mackey, Mehl and Mehrmann, 2006). In particular, we identify the Hermitian linearizations in DL(P) that preserve the sign characteristic of P(λ). We also provide a description of the sign characteristic of the Hermitian linearizations of P(λ) in the family of generalized Fiedler pencils with repetition (Bueno, Dopico, Furtado and Rychnovsky, 2015).

Linearizations of Hermitian Matrix Polynomials Preserving the Sign Characteristic

The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such linearizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In this context, Hermitian matrix polynomials are one of the most important classes of matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic is a set of signs attached to these real eigenvalues which is crucial for determining the behavior of systems described by Hermitian matrix polynomials and, therefore, it is desirable to develop linearizations that preserve the sign characteristic of these polynomials, but, at present, only one such linearization is known. In this paper, we present a complete characterization of all the Hermitian strong linearizations that preserve the sign characteristic of ...

On the sign characteristics of Hermitian matrix polynomials

The sign characteristics of Hermitian matrix polynomials are discussed, and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity. The concept of sign characteristic arises in different forms in many scientific fields, and is essential for the stability analysis in Hamiltonian systems or the perturbation behavior of eigenvalues under structured perturbations. We extend classical results by Gohberg, Lancaster, and Rodman to the case of infinite eigenvalues. We derive a systematic approach, studying how sign characteristics behave after an analytic change of variables, including the important special case of Möbius transformations, and we prove a signature constraint theorem. We also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree Hermitian matrix polynomials, while it may change for odd degree matrix polynomials. We argue that the non-uniformity can be resolved by introducing an extra zero leading matrix coefficient.

<inline-formula> <tex-math notation="TeX">${\cal H}_{\infty}$</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">${\cal H}_{2}$</tex-math></inline-formula> Norms of 2-D Mixed Continuous-Discrete-Time Systems via Rationally-Dependent Complex Lyapunov Functions

This paper addresses the problem of determining the ${\cal H}_{\infty}$ and ${\cal H}_{2}$ norms of 2-D mixed continuous-discrete-time systems. The first contribution is to propose a novel approach based on the use of complex Lyapunov functions with even rational parametric dependence, which searches for upper bounds on the sought norms via linear matrix inequalities (LMIs). The second contribution is to show that the upper bounds provided are nonconservative by using Lyapunov functions in the chosen class with sufficiently large degree. The third contribution is to provide conditions for establishing the tightness of the upper bounds. The fourth contribution is to show how the numerical complexity of the proposed approach can be significantly reduced by proposing a new necessary and sufficient LMI condition for establishing positive semidefiniteness of even Hermitian matrix polynomials. This result is also exploited to derive an improved necessary and sufficient LMI condition for establishing exponential stability of 2-D mixed continuous-discrete-time systems. Some numerical examples illustrate the proposed approach. It is worth remarking that nonconservative LMI methods for determining the ${\cal H}_{\infty}$ and ${\cal H}_{2}$ norms of 2-D mixed continuous-discrete-time systems have not been proposed yet in the literature.

Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Hermitian and Related Structures

We derive a formula for the backward error of a complex number $\lambda$ when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, $*$-even, and $*$-odd. Numerical experiments suggest that in many cases there is a significant difference between the backward errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations.

Instability indices for matrix polynomials

There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directions for the matrix coefficients which are Hermitian to the total number of unstable eigenvalues for the polynomial. Herein we extend the theory to ⋆-even matrix polynomials of any finite degree. In particular, unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two. We also consider Hermitian matrix polynomials, and derive an index which counts the number of eigenvalues with nonpositive imaginary part. The results are refined if we consider the Hermitian matrix polynomial to be a perturbation of a ⋆-even polynomials; however, this refinement requires additional assumptions on the matrix coefficients.

Inverse problems for Hermitian quadratic matrix polynomials

We consider the factorization of Hermitian quadratic matrix polynomials with nonsingular leading coefficient, with special emphasis on the case of real symmetric systems. It is assumed that the quadratic has a nonsingular leading coefficient (rather than the more familiar positive definite hypothesis). Theorems 3 and 8 are the central results showing how to complete a quadratic matrix function when half of the spectral data is specified. Subsequent results give more detailed information on classical problems in which the leading coefficient is positive definite. They concern the distribution of the two distinctive types of real eigenvalues. The theory is illustrated with an informative set of numerical examples.

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.

Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification

The spectral properties of Hermitian matrix polynomials with real eigenvalues have been extensively studied, through classes such as the definite or definitizable pencils, definite, hyperbolic, or quasihyperbolic matrix polynomials, and overdamped or gyroscopically stabilized quadratics. We give a unified treatment of these and related classes that uses the eigenvalue type (or sign characteristic) as a common thread. Equivalent conditions are given for each class in a consistent format. We show that these classes form a hierarchy, all of which are contained in the new class of quasidefinite matrix polynomials. As well as collecting and unifying existing results, we make several new contributions. We propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line. By analyzing their effect on eigenvalue type, we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient, which is important for treating definite and quasidefinite polynomials. We also give a sufficient and necessary condition for a quasihyperbolic matrix polynomial to be strictly isospectral to a real diagonal quasihyperbolic matrix polynomial of the same degree, and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial, thereby identifying an important new class of diagonalizable matrix polynomials.

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 − A 2 , I − A 2 and A − A 3 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.

Hermitian quadratic matrix polynomials: Solvents and inverse problems

A monic quadratic Hermitian matrix polynomial L(λ) can be factorized into a product of two linear matrix polynomials, say L(λ)=(Iλ-S)(Iλ-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.

Detecting hyperbolic and definite matrix polynomials

Hyperbolic or more generally definite matrix polynomials are important classes of Hermitian matrix polynomials. They allow for a definite linearization and can therefore be solved by a standard algorithm for Hermitian matrices. They have only real eigenvalues which can be characterized as minmax and maxmin values of Rayleigh functionals, but there is no easy way to test if a given polynomial is hyperbolic or definite or not. Taking advantage of the safeguarded iteration which converges globally and monotonically to extreme eigenvalues we obtain an efficient algorithm that identifies hyperbolic or definite polynomials and enables the transformation to an equivalent definite linear pencil. Numerical examples demonstrate the efficiency of the approach.

Definite Matrix Polynomials and their Linearization by Definite Pencils

Hyperbolic matrix polynomials are an important class of Hermitian matrix polynomials that contain overdamped quadratics as a special case. They share with definite pencils the spectral property that their eigenvalues are real and semisimple. We extend the definition of hyperbolic matrix polynomial in a way that relaxes the requirement of definiteness of the leading coefficient matrix, yielding what we call definite polynomials. We show that this class of polynomials has an elegant characterization in terms of definiteness intervals on the extended real line and that it includes definite pencils as a special case. A fundamental question is whether a definite matrix polynomial $P$ can be linearized in a structure-preserving way. We show that the answer to this question is affirmative: $P$ is definite if and only if it has a definite linearization in $\mathbb{H}(P)$, a certain vector space of Hermitian pencils; and for definite $P$ we give a complete characterization of all the linearizations in $\mathbb{H}(P)$ that are definite. For the important special case of quadratics, we show how a definite quadratic polynomial can be transformed into a definite linearization with a positive definite leading coefficient matrix—a form that is particularly attractive numerically.

Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems

Hyperbolic quadratic matrix polynomials $Q(\lambda) = \lambda^2 A + \lambda B + C$ are an important class of Hermitian matrix polynomials with real eigenvalues, among which the overdamped quadratics are those with nonpositive eigenvalues. Neither the definition of overdamped nor any of the standard characterizations provides an efficient way to test if a given Q has this property. We show that a quadratically convergent matrix iteration based on cyclic reduction, previously studied by Guo and Lancaster, provides necessary and sufficient conditions for Q to be overdamped. For weakly overdamped Q the iteration is shown to be generically linearly convergent with constant at worst $1/2$, which implies that the convergence of the iteration is reasonably fast in almost all cases of practical interest. We show that the matrix iteration can be implemented in such a way that when overdamping is detected a scalar $\mu<0$ is provided that lies in the gap between the n largest and n smallest eigenvalues of the $n \times n$ quadratic eigenvalue problem (QEP) $Q(\lambda)x = 0$. Once such a $\mu$ is known, the QEP can be solved by linearizing to a definite pencil that can be reduced, using already available Cholesky factorizations, to a standard Hermitian eigenproblem. By incorporating an initial preprocessing stage that shifts a hyperbolic Q so that it is overdamped, we obtain an efficient algorithm that identifies and solves a hyperbolic or overdamped QEP maintaining symmetry throughout and guaranteeing real computed eigenvalues.

Model-updating for self-adjoint quadratic eigenvalue problems

This paper concerns quadratic matrix functions of the form L(λ)=Mλ2+Dλ+K where M,D,K are Hermitian n×n matrices with M>0. It is shown how new systems of the same type can be generated with some eigenvalues and/or eigenvectors updated and this is accomplished without “spill-over” (i.e. other spectral data remain undisturbed). Furthermore, symmetry is preserved. The methods also apply for Hermitian matrix polynomials of higher degree.

Sorted Spectral Factorization of Matrix Polynomials in MIMO Communications

Sorted spectral factorization of matrix polynomials is studied. Such type of factoring Hermitian matrix polynomials is the key step in calculating the optimum receive filter matrices in spatial/temporal decision-feedback equalization as well as the optimum transmit filter matrices in spatial/temporal Tomlinson-Harashima-type precoding schemes. Contrary to other approaches, we inherently consider asymptotic rather than finite-length results for transmission over MIMO channels with intersymbol interference. It is shown how the different types of factorizations can be transformed onto a prototype factorization task, which in turn can be solved by first performing an unsorted factorization and then determining the optimal processing order. An easy-to-use iterative algorithm for unsorted spectral factorization and the adjustment of the optimized order in DFE and precoding are explained. Numerical simulations cover the impact of sorted and unsorted spectral factorization on the performance of DFE and precoding schemes.

Definite triples of Hermitian matrices and matrix polynomials

Let A, B and C be three n× n nonzero Hermitian matrices. The triple ( A, B, C) is called definite if the convex hull of the joint numerical range F(A,B,C)={(x ∗Ax,x ∗Bx,x ∗Cx)∈ R 3 : x∈ C n,x ∗x=1} does not contain (0,0,0). If the triple ( A, B, C) is nondefinite, then the numerical ranges of the matrix polynomials Q(λ)=Aλ ν 3 +Bλ ν 2 +Cλ ν 1 (ν 3>ν 2>ν 1⩾0) and L(λ)=Aλ ξ 2 +(B+ i C)λ ξ 1 (ξ 2>ξ 1⩾0) coincide with the whole complex plane, providing no information. As a consequence, it is of particular interest to characterize a definite triple ( A, B, C) and find the distance between (0,0,0) and the boundary of F( A, B, C). The distance between a nondefinite triple ( A, B, C) and the “nearest” definite triples with specified properties is also investigated. Moreover, applications of definite triples on matrix polynomials of special interest are presented.