Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence

We provide a solution to the problem of simultaneous \(diagonalization\) \(via\) \(congruence\) of a given set of \(m\) complex symmetric \(n\times n\) matrices \(\{A_{1},\ldots,A_{m}\}\), by showing that it can be reduced to a possibly lower-dimensional problem where the question is rephrased in te...

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Bibliographic Details
Published in:arXiv.org 2021-02
Main Authors: Bustamante, Miguel D, Mellon, Pauline, Velasco, M Victoria
Format: Article
Language:English
Subjects:
Algorithms
Mathematical analysis
Matrix methods
Optimization
Online Access:Get full text
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Summary:We provide a solution to the problem of simultaneous \(diagonalization\) \(via\) \(congruence\) of a given set of \(m\) complex symmetric \(n\times n\) matrices \(\{A_{1},\ldots,A_{m}\}\), by showing that it can be reduced to a possibly lower-dimensional problem where the question is rephrased in terms of the classical problem of simultaneous \(diagonalization\) \(via\) \(similarity\) of a new related set of matrices. We provide a procedure to determine in a finite number of steps whether or not a set of matrices is simultaneously diagonalizable by congruence. This solves a long standing problem in the complex case.
ISSN:2331-8422
DOI:10.48550/arxiv.1908.04228