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Invariance properties of a triple matrix product involving generalized inverses. (English) Zbl 1105.15005
Some invariance properties of the product $AXC$ are studied, where $A, B, C$ are complex valued matrices of appropriate dimensions and $X$ is a generalized inverse of $B$. A matrix $X$ is called an $\{i,j,\dots,k\}$-inverse of $B$ and it is denoted by $B^{(i,j,\dots,k)}$ if $X$ satisfies the equations $(i), (j),\dots, (k)$ among the four Penrose equations $$(1)\ BXB=B\quad (2) XBX=X\quad (3)\ (BX)^{*}=BX\quad (4)\ (XB)^{*}=XB.$$ If $X=B^{(1,\dots)}$ then $X$ is also called a generalized inverse of $B$. In the case $X=B^{(1)}$ some known results are reviewed or generalized, results concerning the invariance of the product $AB^{(1)}C$, the range inclusion invariance and the rank invariance. Necessary and sufficient conditions are obtained for similar invariances in the cases $X=B^{(1,2)}, X=B^{(1,3)}$ and $X=B^{(1,4)}$. The cases $X=B^{(1,2,3)}$ and $X=B^{(1,3,4)}$ are discussed, as well as some miscellaneous invariance results, involving eigenvalues or the trace. The connection between the invariance properties and the concept of extremal rank of matrices is emphasized.

15A09Matrix inversion, generalized inverses
15A03Vector spaces, linear dependence, rank
Full Text: DOI
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