Principal Submatrices, Geometric Multiplicities, and Structured Eigenvectors

It is a straightforward matrix calculation that if λ is an eigenvalue of A, x an associated eigenvector and α the set of positions in which x has nonzero entries, then λ is also an eigenvalue of the submatrix of A that lies in the rows and columns indexed by α. A converse is presented that is the most general possible in terms of the data we use. Several corollaries are obtained by applying the main result to normal and Hermitian matrices. These corollaries lead to results concerning the case of equality in the interlacing inequalities for Hermitian matrices, and to the problem of the relationship among eigenvalue multiplicities in various principal submatrices

A matricial proof of the symmetric exchange axiom for eigenvalues of principal submatrices of a complex Hermitian matrix

In [C.R. Johnson, B. Kroschel, M. Omladic, Eigenvalue multiplicities in principal submatrices, Linear Algebra Appl. 390 (2004)111-120] a result constraining the eigenvalues of principal submatrices of complex Hermitian matrices, based upon matroid theory, was given. Here we give a matricial proof of this result which also enables us to find a generalization of the original result. (C) 2013 Elsevier Inc. All rights reserved

The change in eigenvalue multiplicity associated with perturbation of a diagonal entry of the matrix

Here we investigate the relation between perturbing the i-th diagonal entry of A 2 Mn(F) and extracting the principal submatrix A(i) from A with respect to the possible changes in multiplicity of a given eigenvalue. A complete description is given and used to both generalize and improve prior work about Hermitian matrices whose graph is a given tree

Eigenstructure of order-one-quasiseparable matrices. Three-term and two-term recurrence relations

AbstractThis paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order-one-quasiseparable matrices. Various recursive relations for characteristic polynomials of their principal submatrices are derived. The cost of evaluating the characteristic polynomial of an N×N matrix and its derivative is only O(N). This leads immediately to several versions of a fast quasiseparable Newton iteration algorithm. In the Hermitian case we extend the Sturm property to the characteristic polynomials of order-one-quasiseparable matrices which yields to several versions of a fast quasiseparable bisection algorithm.Conditions guaranteeing that an eigenvalue of a order-one-quasiseparable matrix is simple are obtained, and an explicit formula for the corresponding eigenvector is derived. The method is further extended to the case when these conditions are not fulfilled. Several particular examples with tridiagonal, (almost) unitary Hessenberg, and Toeplitz matrices are considered.The algorithms are based on new three-term and two-term recurrence relations for the characteristic polynomials of principal submatrices of order-one-quasiseparable matrices R. It turns out that the latter new class of polynomials generalizes and includes two classical families: (i) polynomials orthogonal on the real line (that play a crucial role in a number of classical algorithms in numerical linear algebra), and (ii) the Szegö polynomials (that play a significant role in signal processing). Moreover, new formulas can be seen as generalizations of the classical three-term recurrence relations for the real orthogonal polynomials and of the two-term recurrence relations for the Szegö polynomials

On a formula of Thompson and McEnteggert for the adjugate matrix

For an eigenvalue lambda(0) of a Hermitian matrix A, the formula of Thompson and McEnteggert gives an explicit expression of the adjugate of lambda I-0 - A, Adj(lambda I-0 - A), in terms of eigenvectors of Afor lambda(0) and all its eigenvalues. In this paper Thompson-McEnteggert's formula is generalized to include any matrix with entries in an arbitrary field. In addition, for any nonsingular matrix A, a formula for the elementary divisors of Adj(A) is provided in terms of those of A. Finally, a generalization of the eigenvalue-eigenvector identity and three applications of the Thompson-McEnteggert's formula are presented.Partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Partially supported by "Ministerio de Economia, Industria y Competitividad (MINECO)" of Spain and "Fondo Europeo de Desarrollo Regional (FEDER)" of EU through grants MTM2017-83624-P and MTM2017-90682-REDT, and by UPV/EHU through grant GIU16/42

Optimal dual frames and frame completions for majorization

In this paper we consider two problems in frame theory. On the one hand, given a set of vectors $\mathcal F$ we describe the spectral and geometrical structure of optimal completions of $\mathcal F$ by a finite family of vectors with prescribed norms, where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Bendetto-Fickus' frame potential. On the other hand, given a fixed frame $\mathcal F$ we describe explicitly the spectral and geometrical structure of optimal frames $\mathcal G$ that are in duality with $\mathcal F$ and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.Comment: 29 pages, with modifications related with the exposition of the materia

Linearizations of rational matrices

Mención Internacional en el título de doctorThis PhD thesis belongs to the area of Numerical Linear Algebra. Specifically, to the numerical solution of the Rational Eigenvalue Problem (REP). This is a type of eigenvalue problem associated with rational matrices, which are matrices whose entries are rational functions. REPs appear directly from applications or as approx imations to arbitrary Nonlinear Eigenvalue Problems (NLEPs). Rational matrices also appear in linear systems and control theory, among other applications. Nowa days, a competitive method for solving REPs is via linearization. This is due to the fact that there exist backward stable and efficient algorithms to solve the linearized problem, which allows to recover the information of the original rational problem. In particular, linearizations transform the REP into a generalized eigenvalue pro blem in such a way that the pole and zero information of the corresponding rational matrix is preserved. To recover the pole and zero information of rational matrices, it is fundamental the notion of polynomial system matrix, introduced by Rosenbrock in 1970, and the fact that rational matrices can always be seen as transfer functions of polynomial system matrices. This thesis addresses different topics regarding the problem of linearizing REPs. On the one hand, one of the main objectives has been to develop a theory of li nearizations of rational matrices to study the properties of the linearizations that have appeared so far in the literature in a general framework. For this purpose, a definition of local linearization of rational matrix is introduced, by developing as starting point the extension of Rosenbrock’s minimal polynomial system matrices to a local scenario. This new theory of local linearizations captures and explains rigor ously the properties of all the different linearizations that have been used from the 1970’s for computing zeros, poles and eigenvalues of rational matrices. In particu lar, this theory has been applied to a number of pencils that have appeared in some influential papers on solving numerically NLEPs through rational approximation. On the other hand, the work has focused on the construction of linearizations of rational matrices taking into account different aspects. In some cases, we focus on preserving particular structures of the corresponding rational matrix in the li nearization. The structures considered are symmetric (Hermitian), skew-symmetric (skew-Hermitian), among others. In other cases, we focus on the direct construc tion of the linearizations from the original representation of the rational matrix. The representations considered are rational matrices expressed as the sum of their polynomial and strictly proper parts, rational matrices written as general trans fer function matrices, and rational matrices expressed by their Laurent expansion around the point at infinity. In addition, we describe the recovery rules of the information of the original rational matrix from the information of the new lineari zations, including in some cases not just the zero and pole information but also the information about the minimal indices. Finally, in this dissertation we tackle one of the most important open problems related to linearizations of rational matrices. That is the analysis of the backward stability for solving REPs by running a backward stable algorithm on a linearization. On this subject, a global backward error analysis has been developed by considering the linearizations in the family of “block Kronecker linearizations”. An analysis of this type had not been developed before in the literature.Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección del profesor Froilán Martínez Dopico y codirección de la profesora Silvia Marcaida Bengoechea. Se contó durante cuatro años con un contrato predoctoral FPI, referencia BES-2016-076744, asociado al proyecto ALGEBRA LINEAL NUMERICA ESTRUCTURADA PARA MATRICES CONSTANTES, POLINOMIALES Y RACIONALES, referencia MTM2015-65798-P, del Ministerio de Economía y Competitividad, y cuyo investigador principal fue Froilán Martínez Dopico. Asociado a este contrato, se contó con una ayuda para realizar parte de este trabajo durante dos es tancias internacionales de investigación. La primera estancia de investigación se realizó del 30 de enero de 2019 hasta el 1 de marzo de 2019 en el Department of Mathematical En gineering, Université catholique de Louvain (Bélgica), bajo la supervisión del profesor Paul Van Dooren. La segunda estancia de investigación se realizó del 15 de septiembre de 2019 hasta el 19 de noviembre de 2019 en el Department of Mathematical Sciences, University of Montana (EEUU), bajo la supervisión del profesor Javier Pérez Alvaro. Dado que la entidad beneficiaria del contrato predoctoral es la UC3M mientras que el otro codirector de tesis, la profesora Silvia Marcaida Bengoechea, pertenece al Departamento de Matemáticas de la Universidad del País Vasco (UPV/EHU), el trabajo con la profesora Silvia Marcaida se reforzó mediante visitas a la UPV/EHU, financiadas por ayudas de la RED temática de Excelencia ALAMA (Algebra Lineal, Análisis Matricial y Aplicaciones) asociadas al los proyectos MTM2015-68805-REDT y MTM2017-90682-REDT.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Ion Zaballa Tejada.- Secretario: Fernando de Terán Vergara.- Vocal: Vanni Noferin

Structured perturbation theory for eigenvalues of symplectic matrices

The problem of computing eigenvalues, eigenvectors, and invariant subspaces of symplectic matrices plays a major role in many applications, in particular in control theory when the focus is on discrete systems. If standard numerical methods for the solution of the symplectic eigenproblem are applied that do not take into account the special symmetry structure of the problem, then not only the existing symmetry in the spectrum of symplectic matrices may be lost in finite precision arithmetic, but more importantly other relevant intrinsic features or invariants may be ignored although they have a major influence in the corresponding computed eigenvalues. The importance of structure-preservation has been acknowledged in the Numerical Linear Algebra community since several decades, and consequently many algorithms have been developed for the symplectic eigenvalue problem that preserve the given structure at each iteration step. The error analysis for such algorithms requires a corresponding stucture-preserving perturbation theory. This is the general framework in which this dissertation can be placed. In this work, a first order perturbation theory for eigenvalues of real or complex Jsymplectic matrices under structure-preserving perturbations is developed. Since the class of symplectic matrices has an underlying multiplicative structure, Lidskii’s classical formulas for small additive perturbations of the form b A = A+εB cannot be applied directly, so a new multiplicative perturbation theory is first developed: given an arbitrary square matrix A, we obtain the leading terms of the asymptotic expansions in the small, real parameter ε of multiplicative perturbations b A(ε) = (I +εB +· · · )A(I +εC +· · · ) of A for arbitrary matrices B and C. The analysis is separated in two complementary cases, depending on whether the unperturbed eigenvalue is zero or not. It is shown that in either case the leading exponents are obtained from the partial multiplicities of the eigenvalue of interest, and the leading coefficients generically involve only appropriately normalized left and right eigenvectors of A associated with that eigenvalue, with no need of generalized eigenvectors. It should be noted that, although initially motivated by the needs for the symplectic case, this multiplicative (unstructured) perturbation theory is of independent interest and stands on its own. After showing that any small structured perturbation bS of a symplectic matrix S can be written as bS = bS(ε) = (I + εB + · · · ) S with Hamiltonian first-order coefficient B, we apply the previously obtained Lidskii-like formulas for multiplicative perturbations to the symplectic case by exploiting the particular connections that symplectic structure induces in the Jordan form between normalized left and right eigenvectors. Special attention is given to eigenvalues on the unit circle, particularly to the exceptional eigenvalues ±1, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under arbitrary ones. Also, several numerical examples are generated Although the approach described above via multiplicative expansions works in most situations, there is a very specific one, the one we call the nongeneric case, which requires a separate, completely different analysis. It corresponds to the case in which, in the absence of structure, the rank of the perturbation would break an odd number of oddsized Jordan blocks corresponding to the eigenvalue either 1 or −1. Since this is not allowed by symplecticity, one among that odd number of Jordan blocks does not break, but increases its size by one becoming an even-sized block. This very special behavior lies outside of the theory developed for what we might call the generic cases, and requires a completely different perturbation analysis, based on Newton diagram techniques like the one performed to obtain the multiplicative expansions. The main difference with the previous expansions is that in this nongeneric case the leading coefficients depend not only on eigenvectors, but also on first generalized Jordan vectors.El problema de calcular autovalores, autovectores y subespacios invariantes de matrices simplécticas juega un papel crucial en muchas aplicaciones, en particular en la Teoría de Control cuando ésta se centra en sistemas discretos. Si para resolver el problema simpléctico de autovalores se emplean métodos numéricos estándar que no tienen en cuenta la simetría especial del problema, entonces no solo se perderá en aritmética finita la simetría natural del espectro de las matrices simplécticas, sino que, aún más importante, podemos estar ignorando otras características o invariantes intrínsecas que tienen una influencia crucial en los correspondientes autovalores calulados. La importancia de preservar la estructura ha sido reconocida por la comunidad del Álgebra Lineal Numérica desde hace varias décadas y, en consecuencia, se han desarrollado diversos algoritmos para el problema simpléctico de autovalores que mantienen la estructura simpléctica en cada paso del proceso iterativo. El análisis de errores de tales algoritmos demanda una teoría de perturbación asociada que también preserve la estructura. Este es el marco general en el que se puede inscribir esta tesis doctoral. En este trabajo se desarrolla una teoría de perturbación de autovalores de matrices J-simplécticas frente a perturbaciones que preservan la simplecticidad de la matriz. Dado que la clase de matrices simplécticas tiene una estructura multiplicativa subyacente, las fórmulas clásicas de Lidskii para perturbaciones aditivas pequeñas de la forma b A = A + εB no se pueden aplicar de manera directa, de modo que desarrollamos una nueva teoría de perturbación multiplicativa: dada cualquier matriz cuadrada A, obtenemos el término director del desarrollo asintótico en el parámetro real (y pequeño) ε de autovalores de perturbaciones multiplicativas b A(ε) = (I + εB + · · · )A(I + εC + · · · ) de A para matrices arbitrarias B y C. El análisis se separa en dos casos complementarios, dependiendo de que el autovalor a perturbar sea nulo o no. Se demuestra que en ambos casos los exponentes directores se obtienen a partir de las multiplicidades parciales del autvalor bajo estudio, y que los coeficientes directores solo involucran genéricamente autovectores derechos e izquierdos adecuadamente normalizados, sin necesidad de autovalor generalizado alguno. Debe señalarse que, aunque inicialmente motivados por la necesidad para el caso simpléctico, esta teoría (no estructurada) de perturbación multiplicativa reviste interés per se independientemente de su aplicación al caso simpléctico. Tras mostrar que cualquier perturbación estructurada peque na bS de una matriz simpléctica S puede escribirse como bS = bS(ε) = (I + εB + · · · ) S con coeficiente de primer orden B Hamiltoniano, aplicamos las fórmulas tipo Lidskii obtenidas para perturbaciones multiplicativas al caso simpléctico, explotando la particular conexión que la estructura simpléctica induce entre los autovectores derechos e izquierdos normalizados por la forma de Jordan. Especial atención se le dedica a los autovalores sobre el círculo unidad, particularmente a los autovalores excepcionales ±1, cuyo comportamiento frente a perturbaciones estructuradas es sabido que difiere muy significativemente del comportamiento frente a perturbaciones arbitrarias. Además, presentamos varios ejemplos numéricos que ilustran (y confirman) los desarrollos asintóticos obtenidos. Aunque el enfoque que acabamos de describir via desarrollos multiplicativos funciona en la mayor parte de las situaciones, hay una muy específica, la que llamamos el caso no-genérico, que requiere de un análisis por separado, completamete distinto del anterior. Corresponde al caso en que, en ausencia de estructura, el rango de la perturbación rompería un número impar de bloques de Jordan de tamaño impar asociados a uno de los autovalores 1 ó −1. Como esto es incompatible con la simplecticidad, uno de entre los bloques de tamaño impar no se rompe, sino que incrementa en uno su dimensión, conviertiéndose en un bloque de Jordn de tamaño par. Este comportamiento tan especial no está explicado por la teoría de lo que podríamos llamar los casos ‘genéricos’ , y requiere de un análisis de perturbación completamente distinto, basado en técnicas del Diagrama de Newton, como el llevado a cabo para obtener los desarrollos multiplicativos. La diferencia principal con los desarrollos anteriores es que en el caso no genérico los coeficientes directores dependen no solo de autovectores, sino también de vectores primeros generalizados de Jordan.Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección del profesor Julio Moro Carreño. Se contó con una beca de la UC3M de ayuda al estudio de máster y posteriormente con un contrato predoctoral de la UC3M. Adicionalmente se recibió ayuda parcial del proyecto de investigación: “Matemáticas e Información Cuántica: de las Álgebras de Operadores al Muestreo Cuántico” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2014-54692-P).Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Luis Alberto Ibort Latre.- Secretario: Rafael Cantó Colomina.- Vocal: Francisco Enrique Velasco Angul