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Relationships between generalized inverses of a matrix and generalized inverses of its rank-one-modifications. (English) Zbl 1060.15010

This paper is related to the known Sherman-Morrison formula. The case of rectangular and singular matrices is analyzed. Different results for rank-one-modifications involving generalized inverses (i.e., \(A^-\) such that \(AA^-A=A\)) or reflexive generalized inverses (i.e., \(A^{=}\) such that \(AA^=A=A\) and \(A^{=}AA^{=}=A^{=}\)) are given. Specifically, relationships between generalized inverses \(A^-\) (or \(A^=\)) and \(M^-\) (or \(M^=\)) are studied being \(M=A+bc^*\) a rank-one-modification. Problems such as: when a given generalized inverse \(A^-\) of \(A\) is simultaneously a generalized inverse of \(M\), or when all generalized inverses of \(A\) are also generalized inverses of \(M\), are solved.

MSC:

15A09 Theory of matrix inversion and generalized inverses
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