Characterization and properties of matrices with generalized symmetry or skew symmetry
In: Linear Algebra and its Applications, Jg. 377 (2004), S. 207-218
Online
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Zugriff:
Let R∈Cn×n be a nontrivial involution; i.e., R=R−1≠±I. We say that A∈Cn×n is R-symmetric (R-skew symmetric) if RAR=A (RAR=−A).There are positive integers r and s with r+s=n and matrices P∈Cn×r and Q∈Cn×s such that P∗P=Ir, Q∗Q=Is, RP=P, and RQ=−Q. We give an explicit representation of an arbitrary R-symmetric matrix A in terms of P and Q, and show that solving Az=w and the eigenvalue problem for A reduce to the corresponding problems for matrices APP∈Cr×r and AQQ∈Cs×s. We also express A−1 in terms of APP−1 and AQQ−1. Under the additional assumption that R∗=R, we show that Moore–Penrose inversion and singular value decomposition of A reduce to the corresponding problems for APP and AQQ. We give similar results for R-skew symmetric matrices. These results are known for the case where R=J=(δi,n−j+1)i,j=1n; however, our proofs are simpler even in this case.We say that A∈Cn×n is R-conjugate if RAR=Ā where R∈Rn×n and R=R−1≠±I. In this case R(A) is R-symmetric and I(A) is R-skew symmetric, so our results provide explicit representations for R-conjugate matrices in terms of P and Q, which are now in Rn×r and Rn×s respectively. We show that solving Az=w, inverting A, and the eigenvalue problem for A reduce to the corresponding problems for a related matrix S∈Rn×n. If RT=R this is also true for Moore–Penrose inversion and singular value decomposition of A.
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Characterization and properties of matrices with generalized symmetry or skew symmetry
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Autor/in / Beteiligte Person: | Trench, William F. |
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Zeitschrift: | Linear Algebra and its Applications, Jg. 377 (2004), S. 207-218 |
Veröffentlichung: | Elsevier BV, 2004 |
Medientyp: | unknown |
ISSN: | 0024-3795 (print) |
DOI: | 10.1016/j.laa.2003.07.013 |
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