A QR-LIKE ALGORITHM FOR THE PALINDROMIC EIGENVALUE PROBLEM
Please always quote using this URN:urn:nbn:de:0296-matheon-3921
- In this paper we develop a QR-like algorithm for the palindromic eigenvalue problem $Ax=\lambda A^\adj x$. We will discuss the two cases that $A^\adj$ denotes the transpose or the conjugate transpose of $A\in\C^{n,n}$. It is shown that this so-called palindromic QR iteration is equivalent to applying the standard QR algorithm to $A^{-\adj}A$. Also the concepts of deflation, shifting, and exploiting the invariance of a Hessenberg-type form are adapted. Moreover, we analyze the problem of reducing a general square matrix to the mentioned Hessenberg-type form and establish analogies to the Hamiltonian eigenvalue problem. Finally, we present concrete Hessenberg-type reduction algorithms for special cases.
Author: | Christian Schröder |
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URN: | urn:nbn:de:0296-matheon-3921 |
Referee: | Volker Mehrmann |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2007/05/23 |
Release Date: | 2007/05/23 |
Tag: | |
Institute: | Technische Universität Berlin |
MSC-Classification: | 15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A18 Eigenvalues, singular values, and eigenvectors |
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A21 Canonical forms, reductions, classification | |
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A22 Matrix pencils [See also 47A56] | |
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A23 Factorization of matrices | |
65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F15 Eigenvalues, eigenvectors | |
Preprint Number: | 388 |