Characterization and properties of matrices with generalized symmetry or skew symmetry

By W. F. Trench. 2004

82 citations recorded by the source corpus.

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=A 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/ed715c8292f88e42c1230b8e0a62ce0693b8fccd

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In Algebra and Number Theory

1. Qing-Wen Wang
2. J. Woude
3. Haixia Chang
4. Zhuo-Heng He

In Computational Theory and Mathematics

1. Qing-Wen Wang
2. J. Woude
3. Haixia Chang
4. Zhuo-Heng He
5. Mehdi Dehghan
6. Masoud Hajarian
7. Ivan Kyrchei

In Electrical and Electronic Engineering

1. J. Woude
2. Qing-Wen Wang
3. Haixia Chang

Citations in the corpus

• Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry (2004, unknown): “Generalizations of centrosymmetric and centroskew matrices have recently been studied in [7, 9, 10, 12, 14 ].”
• A system of real quaternion matrix equations with applications (2009, methodology): “Criteria for a matrix to be P-symmetric (or P-skewsymmetric) and (P,Q )symmetric (or (P,Q )-skewsymmetric) were derived by Trench in [27,30,31].”
• Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices (2004, unknown): “If m = n and R = S, our results apply to R symmetric and R-skew symmetric matrices, which are discussed in [7].”