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On some characterizations of the ”star” partial ordering for matrices and rank subtractivity. (English) Zbl 0603.15001
Authors’ summary: The main result is that Drazin’s ”star partial ordering \(A\leq_{*}B\) holds if and only if \(A\ll B\) and \(B^{†}- A^{†}=(B-A)^{†}\), where \(A\leq_{*}B\) is defined by \(A^*A=A^*B\) and \(AA^*=BA^*\), and where \(A\ll B\) denotes rank subtractivity. Here A and B are \(m\times n\) complex matrices and the superscript \(†\) denotes the Moore-Penrose inverse. Several other characterizations of \(A\leq_{*}B\) are given, with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger ”star” order; a key tool is a new canonical form of rank subtractivity. Connections with simultaneous singular-value decompositions, Schur complements, and idempotent matrices are also mentioned.
Reviewer: S.L.Campbell

MSC:
15A09 Theory of matrix inversion and generalized inverses
06F25 Ordered rings, algebras, modules
15A03 Vector spaces, linear dependence, rank, lineability
15A18 Eigenvalues, singular values, and eigenvectors
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