Spectral conditioning and pseudospectral growth

Linear Algebra and its Applications, 2012

The interplay between the algebraic and analytic properties of a matrix and the geometric properties of its pseudospectrum is investigated. It is shown that one can characterize Hermitian matrices, positive semi-definite matrices, orthogonal projections, unitary matrices, etc. in terms of the pseudospectrum. Also, characterizations are given to maps on matrices leaving invariant the pseudospectrum of the sum, difference, or product of matrix pairs. It is shown that such a map is always a unitary similarity transform followed by some simple operations such as adding a constant matrix, taking the matrix transpose, or multiplying by a scalar in {1, −1}.

Numerische Mathematik, 1991

In this paper we investigate the set of eigenvalues of a perturbed matrix A + ∆ ∈ R n×n where A is given and ∆ ∈ R n×n , ∆ < ρ is arbitrary. We determine a lower bound for this spectral value set which is exact for normal matrices A with well separated eigenvalues. We also investigate the behaviour of the spectral value set under similarity transformations. The results are then applied to stability radii which measure the distance of a matrix A from the set of matrices having at least one eigenvalue in a given closed instability domain C b ⊂ C.

2015

Numerical range and pseudospectra of a matrix play an important role in different areas and have several applications in different domains. The numerical range gives an estimate to the location of the pseudospectra, the present work proposes some properties of pseudospectra of a matrix and provides a connection between numerical range and pseudospectra of a matrix. Some upper bounds for pseudospectra of a matrix are given. Also we use the pseudospectra to solve a problem concerning the matrices which are nearly commuting.

2004

Abstract. Transfer functions have been shown to provide monotonic approximations to the resolvent 2-norm of A, R (z)=(A− zI)− 1, when associated with a sequence of nested spaces. This paper addresses the open question of the effectiveness of the transfer function scheme for the computation of the pseudospectrum of large matrices.

Cubo (Temuco)

The objective of the study was to investigate a new notion of generalized trace pseudospectrum for an matrix pencils. In particular, we prove many new interesting properties of the generalized trace pseudo-spectrum. In addition, we show an analogue of the spectral mapping theorem for the generalized trace pseudo-spectrum in the matrix algebra. RESUMEN El objetivo de este estudio es investigar una nueva noción de pseudo-espectro traza generalizado para pinceles de matrices. En particular, demostramos variadas propiedades nuevas e interesantes del pseudo-espectro traza generalizado. Adicionalmente, mostramos un análogo del teorema espectral de aplicaciones para el pseudo-espectro traza generalizado en elálgebra de matrices.

Journal of Computational and Applied Mathematics

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Boletim da Sociedade Paranaense de Matemática

We work with the notion of trace pseudospectra for an element in the matrix algebra. Many new interesting properties of the trace pseudospectrum have been discovered. In addition, we show an analogue of the spectral mapping theorem for trace pseudospectrum in the matrix algebra. Among other things, we illustrated the applicability of this concepts by a considerable number of examples.

2011

In this paper we develop and apply methods for the spectral analysis of non-self-adjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a major application to illustrate our methods we focus on the &quot;hopping sign model&quot; introduced by J.Feinberg and A.Zee (Phys. Rev. E 59 (1999), 6433-6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ± 1&#39;s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and p-norm -pseudospectra (&gt;0, p∈ [1,∞]) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained i...

2016

We present a new iterative algorithm to compute the pseudospectral abscissa for a class of nonlinear eigenvalue problems, which includes the delay and the polynomial eigenvalue problem. Two methods are developed that exploit real valued matrix perturbations, for the Frobenius and the spectral norm. In both cases, it is proved that the pseudospectral abscissa can be obtained when restricting to at most rank-2 perturbations and that the critical perturbations on the coefficient share their column and row spaces. This is a surprising property, giving that the combined additive perturbation on the characteristic matrix is highly structured. The proposed iterative algorithms can be interpreted as the discretization of a gradient ow, where properties of critical perturbations are fully exploited. Even though the main contributions concern the nonlinear eigenvalue problem, the paper contributes to the special case of the standard eigenvalue problem in the following way. For the Frobenius norm perturbations we provide differential equations generating a path in the space of perturbations of rank smaller or equal to two (instead of rank exactly two in the present literature), allowing us to treat the case of complex and real rightmost eigenvalues simultaneously. For the spectral norm case we derive explicit optimality conditions in terms of the compact singular value decomposition. Finally we provide two extensions for the Frobenius norm pseudospectrum. The first one consists of using a combined measure on the perturbations, allowing us to assess from the critical perturbations which coefficient matrices the pseudospectral abscissa is most sensitive to. The second extension consists of considering structured perturbations on the coefficient matrices. In this way our algorithm exploits structure at three levels: the structure of the nonlinear eigenvalue problem (instead of studying unstructured pseudospectra of some linearization of the eigenvalue problem), the property that perturbations are assumed to be real valued, and additional structure on the perturbations of individual matrices. For this case we show that the pseudospectral abscissa cannot always be reached when restricting to perturbations of maximum size.

Linear Algebra and its Applications, 2020

We approximate the pseudospectra of a block triangular matrix with two blocks on the diagonal in terms of the pseudospectra of its diagonal blocks.