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Taussky' s theorem, symmetrizability and modal analysis revisted

Taussky' s theorem, symmetrizability and modal analysis revisted
Taussky' s theorem, symmetrizability and modal analysis revisted
This paper is concerned with symmetrization and diagonalization of real matrices and their implications for the dynamics of linear, second-order systems governed by equations of motion having asymmetric coefficient matrices. Results in the light of Taussky's theorem are presented. The connection of the symmetrizers with the eigenvalue problem is brought out. An alternative proof of Taussky's theorem for real matrices is presented. Diagonalization of two real symmetric (but not necessarily positive-definite) matrices is discussed in the context of undamped non-gyroscopic systems. A commutator of two matrices with respect to a given third matrix is defined; this commutator is found to play an interesting role in deciding simultaneous diagonalizability of two or three matrices. Errors in a few previously known results are brought out. Pseudo-conservative systems are studied and their connection with the so-called 'symmetrizable systems' is critically examined. Results for modal analysis of general non-conservative systems are presented. Illustrative examples are given.
eigenvalue, normal, nodes, diagonalization
1364-5021
2455-2480
Bhaskar, Atul
d4122e7c-5bf3-415f-9846-5b0fed645f3e
Bhaskar, Atul
d4122e7c-5bf3-415f-9846-5b0fed645f3e

Bhaskar, Atul (2001) Taussky' s theorem, symmetrizability and modal analysis revisted. Proceedings of the Royal Society A, 457 (2014), 2455-2480. (doi:10.1098/rspa.2001.0820).

Record type: Article

Abstract

This paper is concerned with symmetrization and diagonalization of real matrices and their implications for the dynamics of linear, second-order systems governed by equations of motion having asymmetric coefficient matrices. Results in the light of Taussky's theorem are presented. The connection of the symmetrizers with the eigenvalue problem is brought out. An alternative proof of Taussky's theorem for real matrices is presented. Diagonalization of two real symmetric (but not necessarily positive-definite) matrices is discussed in the context of undamped non-gyroscopic systems. A commutator of two matrices with respect to a given third matrix is defined; this commutator is found to play an interesting role in deciding simultaneous diagonalizability of two or three matrices. Errors in a few previously known results are brought out. Pseudo-conservative systems are studied and their connection with the so-called 'symmetrizable systems' is critically examined. Results for modal analysis of general non-conservative systems are presented. Illustrative examples are given.

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Published date: 2001
Keywords: eigenvalue, normal, nodes, diagonalization

Identifiers

Local EPrints ID: 21860
URI: https://eprints.soton.ac.uk/id/eprint/21860
ISSN: 1364-5021
PURE UUID: 2b17cfd3-6b31-41d9-bce3-afe326e70ec8

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Date deposited: 16 Mar 2006
Last modified: 19 Jul 2019 19:12

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