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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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