On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices
The singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, 1σ{1 \over \sigma }). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left an...
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Online Access: | https://doi.org/10.1515/spma-2020-0001 |
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doaj-9794f7b87ae84ff8808886712eeab6762021-10-02T18:54:20ZengDe GruyterSpecial Matrices2300-74512020-01-018111310.1515/spma-2020-0001spma-2020-0001On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matricesFaßbender Heike0Halwaß Martin1Institut Computational Mathematics/AG Numerik, TU Braunschweig, Universitätsplatz 2, 38106Braunschweig, GermanyKampstraße 7, 17121Loitz, GermanyThe singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, 1σ{1 \over \sigma }). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left and right singular vectors or by themselves. The link between the left and right singular vectors is used to reformulate the singular value decomposition (SVD) of an involutory matrix as an eigendecomposition. This displays an interesting relation between the singular values of an involutory matrix and its eigenvalues. Similar observations hold for the SVD, the singular values and the coneigenvalues of (skew-)coninvolutory matrices.https://doi.org/10.1515/spma-2020-0001singular value decomposition(skew-)involutory matrix(skew-)coninvolutoryconsimilarity15a2365f99 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Faßbender Heike Halwaß Martin |
spellingShingle |
Faßbender Heike Halwaß Martin On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices Special Matrices singular value decomposition (skew-)involutory matrix (skew-)coninvolutory consimilarity 15a23 65f99 |
author_facet |
Faßbender Heike Halwaß Martin |
author_sort |
Faßbender Heike |
title |
On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices |
title_short |
On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices |
title_full |
On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices |
title_fullStr |
On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices |
title_full_unstemmed |
On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices |
title_sort |
on the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices |
publisher |
De Gruyter |
series |
Special Matrices |
issn |
2300-7451 |
publishDate |
2020-01-01 |
description |
The singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, 1σ{1 \over \sigma }). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left and right singular vectors or by themselves. The link between the left and right singular vectors is used to reformulate the singular value decomposition (SVD) of an involutory matrix as an eigendecomposition. This displays an interesting relation between the singular values of an involutory matrix and its eigenvalues. Similar observations hold for the SVD, the singular values and the coneigenvalues of (skew-)coninvolutory matrices. |
topic |
singular value decomposition (skew-)involutory matrix (skew-)coninvolutory consimilarity 15a23 65f99 |
url |
https://doi.org/10.1515/spma-2020-0001 |
work_keys_str_mv |
AT faßbenderheike onthesingularvaluedecompositionofskewinvolutoryandskewconinvolutorymatrices AT halwaßmartin onthesingularvaluedecompositionofskewinvolutoryandskewconinvolutorymatrices |
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1716848585909207040 |