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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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