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

Victor A. Bovdi; Tatiana G. Gerasimova; Mohamed A. Salim; Vladimir V. Sergeichuk

doi:10.1016/j.laa.2017.12.013

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

Victor A. Bovdi, Tatiana G. Gerasimova, Mohamed A. Salim, Vladimir V. Sergeichuk

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Let (A,B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum (A__,B__)⊕(A₁,B₁)⊕…⊕(A_t,B_t) that is congruent to (A,B), in which (A__,B__) is a pair of nonsingular matrices and (A₁,B₁),…,(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.

Original language	English
Pages (from-to)	17-30
Number of pages	14
Journal	Linear Algebra and Its Applications
Volume	543
DOIs	https://doi.org/10.1016/j.laa.2017.12.013
Publication status	Published - Apr 15 2018

Keywords

Canonical form
Pair of skew-symmetric matrices
Regularization decomposition

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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title = "Reduction of a pair of skew-symmetric matrices to its canonical form under congruence",

abstract = "Let (A,B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum (A__,B__)⊕(A1,B1)⊕…⊕(At,Bt) that is congruent to (A,B), in which (A__,B__) is a pair of nonsingular matrices and (A1,B1),…,(At,Bt) 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.",

keywords = "Canonical form, Pair of skew-symmetric matrices, Regularization decomposition",

author = "Bovdi, {Victor A.} and Gerasimova, {Tatiana G.} and Salim, {Mohamed A.} and Sergeichuk, {Vladimir V.}",

note = "Funding Information: The work was supported in part by the UAEU UPAR grants G00001922 and G00002160 . Funding Information: The work was supported in part by the UAEU UPAR grants G00001922 and G00002160. Publisher Copyright: {\textcopyright} 2017 Elsevier Inc.",

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T1 - Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

AU - Bovdi, Victor A.

AU - Gerasimova, Tatiana G.

AU - Salim, Mohamed A.

AU - Sergeichuk, Vladimir V.

N1 - Funding Information: The work was supported in part by the UAEU UPAR grants G00001922 and G00002160 . Funding Information: The work was supported in part by the UAEU UPAR grants G00001922 and G00002160. Publisher Copyright: © 2017 Elsevier Inc.

PY - 2018/4/15

Y1 - 2018/4/15

N2 - Let (A,B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum (A__,B__)⊕(A1,B1)⊕…⊕(At,Bt) that is congruent to (A,B), in which (A__,B__) is a pair of nonsingular matrices and (A1,B1),…,(At,Bt) 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.

AB - Let (A,B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum (A__,B__)⊕(A1,B1)⊕…⊕(At,Bt) that is congruent to (A,B), in which (A__,B__) is a pair of nonsingular matrices and (A1,B1),…,(At,Bt) 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.

KW - Canonical form

KW - Pair of skew-symmetric matrices

KW - Regularization decomposition

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DO - 10.1016/j.laa.2017.12.013

M3 - Article

AN - SCOPUS:85038825169

SN - 0024-3795

VL - 543

SP - 17

EP - 30

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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