Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

Let \((A,B)\) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum \[ (\underline{\underline A},\underline{\underline B})\oplus (A_1,B_1)\oplus\dots\oplus(A_t,B_t) \] that is congruent to \((A,B)\), in which \((\underline{\underl...

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Bibliographic Details
Published in:arXiv.org 2017-12
Main Authors: Bovdi, V A, Gerasimova, T G, Salim, M A, Sergeichuk, V V
Format: Article
Language:English
Subjects:
Algorithms
Decomposition
Mathematical analysis
Matrix methods
Regularization
Online Access:Get full text
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Summary:Let \((A,B)\) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum \[ (\underline{\underline A},\underline{\underline B})\oplus (A_1,B_1)\oplus\dots\oplus(A_t,B_t) \] that is congruent to \((A,B)\), in which \((\underline{\underline A},\underline{\underline B})\) is a pair of nonsingular matrices and \((A_1,B_1),\) \(\dots,\) \((A_t,B_t)\) are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of \((A,B)\) under congruence over an algebraically closed field of characteristic not 2.
ISSN:2331-8422
DOI:10.48550/arxiv.1712.08729