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More on extremal ranks of the matrix expressions \(A-BX \pm X^{*}B^{*}\) with statistical applications. (English) Zbl 1212.15029
Summary: Through a Hermitian-type (skew-Hermitian-type) singular value decomposition for a pair of matrices \((A, B)\) introduced by H. Zha [Linear Algebra Appl. 240, 199–205 (1996; Zbl 0923.15007)], where \(A\) is Hermitian (skew-Hermitian), we show how to find a Hermitian (skew-Hermitian) matrix X such that the matrix expressions \(A-BX \pm X^{*}B^{*}\) achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations \(BX \pm X^{*}B^{*}=A\), we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model \(\{y, X\beta , \sigma ^{2}V\}\), we discuss the existence of a symmetric matrix \(G\) such that \(Gy\) is the weighted least-squares estimator and the best linear unbiased estimator of \(X\beta \), respectively.

MSC:
15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
15A22 Matrix pencils
62J05 Linear regression; mixed models
65F20 Numerical solutions to overdetermined systems, pseudoinverses
62F10 Point estimation
15A03 Vector spaces, linear dependence, rank, lineability
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