Characterization and Properties of (R, S)-Symmetric, (R, S)-Skew Symmetric, and (R, S)-Conjugate Matrices
نویسنده
چکیده
SIAM J. Matrix Anal Appl. 26 (2005) 748–757 Abstract. Let R ∈ C and S ∈ C be nontrivial involutions; i.e., R = R 6= ±Im and S = S 6= ±In. We say that A ∈ C is (R,S)-symmetric ((R,S)-skew symmetric) if RAS = A (RAS = −A). We give an explicit representation of an arbitrary (R,S)-symmetric matrix A in terms of matrices P andQ associatedwith R and U and V associatedwith S. If R = R, then the least squares problem for A can be solved by solving the independent least squares problems for APU = P AU ∈ C and AQV = Q ∗AV ∈ Cs×`, where r + s = m and k + ` = n. If, in addition, either rank(A) = n or S = S, then A can be expressed in terms of A † PU and A † QV . If R = R and S = S, then a singular value decomposition of A can obtained from singular value decompositions of APU and AQV . Similar results hold for (R,S)-skew symmetric matrices. We say that A ∈ Cm×n is R-conjugate if RAS = R, where R ∈ Rm×m and S ∈ Rn×n, R = R 6= ±Im, and S = S 6= ±In. In this case <(A) is (R,S)-symmetric and =(A) is (R,S)skew symmetric, so our results provide explicit representations for (R,S)-conjugate matrices. If R = R the least squares problem for the complex matrix A reduces to two least squares problems for a real matrix K. If, in addition, either rank(A) = n or ST = S, then A† can be obtained from K. If both R = R and S = S, a singular value decomposition of A can be obtained from a singular value decomposition of K.
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ورودعنوان ژورنال:
- SIAM J. Matrix Analysis Applications
دوره 26 شماره
صفحات -
تاریخ انتشار 2005