scholarly journals Structured eigenvalue backward errors of matrix polynomials

2015 ◽  
Author(s):  
Punit Sharma ◽  
Shreemayee Bora ◽  
Michael Karow ◽  
Christian Mehl
Keyword(s):  
Numerical Results ◽  
Real Matrix ◽  
Structure Preserving ◽  
Matrix Polynomials ◽  
Significant Difference ◽  
Matrix Pencils ◽  
Backward Errors

In this poster, we briefly present some results on eigenvalue backward errors of matrix pencils and polynomials under structure preserving perturbations. We also present eigenvalue backward errors of real matrix pencils with respect to real perturbations that also preserve certain structures like symmetric, T-alternating and T-palindromic. Numerical results show that there is a significant difference between the backward errors with respect to perturbations that preserve structures and those with respect to arbitrary perturbations.

scholarly journals Structured eigenvalue/eigenvector backward errors of matrix pencils arising in optimal control

2018 ◽  
Vol 34 ◽  
pp. 526-560
Author(s):  
Christian Mehl ◽  
Volker Mehrmann ◽  
Punit Sharma
Keyword(s):  
Optimal Control ◽  
Block Structure ◽  
Special Structure ◽  
Structure Preserving ◽  
Matrix Pencils ◽  
Symmetry Structure ◽  
Backward Errors

Eigenvalue and eigenpair backward errors are computed for matrix pencils arising in optimal control. In particular, formulas for backward errors are developed that are obtained under block-structure-preserving and symmetry-structure-preserving perturbations. It is shown that these eigenvalue and eigenpair backward errors are sometimes significantly larger than the corresponding backward errors that are obtained under perturbations that ignore the special structure of the pencil.

scholarly journals Sign Characteristics of Regular Hermitian Matrix Pencils under Generic Rank-1 and Rank-2 Perturbations

2015 ◽  
Vol 30 ◽  
pp. 760-794 ◽  
Author(s):  
Leonhard Batzke
Keyword(s):  
Canonical Form ◽  
Hermitian Matrix ◽  
Structure Preserving ◽  
Spectral Behavior ◽  
Generic Rank ◽  
Jordan Structure ◽  
Matrix Pencils ◽  
Rank 2 ◽  
Infinite Blocks

The spectral behavior of regular Hermitian matrix pencils is examined under certain structure-preserving rank-1 and rank-2 perturbations. Since Hermitian pencils have signs attached to real (and infinite) blocks in canonical form, it is not only the Jordan structure but also this so-called sign characteristic that needs to be examined under perturbation. The observed effects are as follows: Under a rank-1 or rank-2 perturbation, generically the largest one or two, respectively, Jordan blocks at each eigenvalue lambda are destroyed, and if lambda is an eigenvalue of the perturbation, also one new block of size one is created at lambda. If lambda is real (or infinite), additionally all signs at lambda but one or two, respectively, that correspond to the destroyed blocks, are preserved under perturbation. Also, if the potential new block of size one is real, its sign is in most cases prescribed to be the sign that is attached to the eigenvalue lambda in the perturbation.

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

2014 ◽  
Vol 35 (2) ◽  
pp. 453-475 ◽  
Author(s):  
Shreemayee Bora ◽  
Michael Karow ◽  
Christian Mehl ◽  
Punit Sharma
Keyword(s):  
Matrix Pencils ◽  
Backward Errors

Structured Backward Errors and Pseudospectra of Structured Matrix Pencils

2009 ◽  
Vol 31 (2) ◽  
pp. 331-359 ◽  
Author(s):  
Bibhas Adhikari ◽  
Rafikul Alam
Keyword(s):  
Structured Matrix ◽  
Matrix Pencils ◽  
Backward Errors

scholarly journals Palindromic quadratization and structure-preserving algorithm for palindromic matrix polynomials of even degree

2011 ◽  
Vol 118 (4) ◽  
pp. 713-735 ◽  
Author(s):  
Tsung-Ming Huang ◽  
Wen-Wei Lin ◽  
Wei-Shuo Su
Keyword(s):  
Structure Preserving ◽  
Matrix Polynomials

scholarly journals Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

2019 ◽  
Vol 35 ◽  
pp. 116-155
Author(s):  
Biswajit Das ◽  
Shreemayee Bora
Keyword(s):  
Eigenvalue Problem ◽  
Matrix Polynomial ◽  
Vector Spaces ◽  
Backward Error ◽  
Rectangular Matrix ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
Stable Manner ◽  
The Matrix ◽  
Matrix Pencils

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

Appendix: Real and complex canonical forms

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Canonical Forms ◽  
Real Matrix ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Matrix Pencils ◽  
Real Matrices

This chapter presents canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are well-known. The chapter first looks at Jordan and Kronecker canonical forms, before turning to real matrix pencils with symmetries. It provides canonical forms for pairs of real matrices, either one of which is symmetric or skewsymmetric, or what is the same, corresponding matrix pencils. Finally, this chapter presents canonical forms of complex matrix pencils with various symmetries, such as complex matrix pencils with symmetries with respect to transposition.

scholarly journals Block Kronecker linearizations of matrix polynomials and their backward errors

2018 ◽  
Vol 140 (2) ◽  
pp. 373-426 ◽  
Author(s):  
Froilán M. Dopico ◽  
Piers W. Lawrence ◽  
Javier Pérez ◽  
Paul Van Dooren
Keyword(s):  
Matrix Polynomials ◽  
Backward Errors
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