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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.

MSC:
15A09Matrix inversion, generalized inverses
15A03Vector spaces, linear dependence, rank
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Full Text: DOI
References:
[1] Baksalary, J. K.: Strong unified-least-squares matrices for a general linear model. Linear algebra appl. 70, 61-65 (1985) · Zbl 0584.62076
[2] Baksalary, J. K.: A new approach to the concept of a strong unified-least squares matrix. Linear algebra appl. 388, 7-15 (2004) · Zbl 1060.15009
[3] Baksalary, J. K.; Kala, R.: Range invariance of certain matrix products. Linear and multilinear algebra 14, 89-96 (1983) · Zbl 0523.15006
[4] Baksalary, J. K.; Markiewicz, A.: Further results on invariance of the eigenvalues of matrix products involving generalized inverses. Linear algebra appl. 237 -- 238, 115-121 (1996) · Zbl 0851.15006
[5] Baksalary, J. K.; Mathew, T.: Rank invariance criterion and its application to the unified theory of least squares. Linear algebra appl. 127, 393-401 (1990) · Zbl 0694.15003
[6] Baksalary, J. K.; Pukkila, T.: A note on invariance of the eigenvalues, singular values, and norms of matrix products involving generalized inverses. Linear algebra appl. 165, 125-130 (1992) · Zbl 0743.15005
[7] Ben-Irael, A.; Greville, T. E.: Generalized inverses. Theory and applications. (2003) · Zbl 1026.15004
[8] Groß, J.: Comment on range invariance of matrix products. Linear and multilinear algebra 41, 157-160 (1996) · Zbl 0871.15003
[9] Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices. Linear and multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003
[10] Penrose, R.: A generalized inverse for matrices. Proc. Cambridge philos. Soc. 51, 406-413 (1955) · Zbl 0065.24603
[11] Rao, C. R.; Mitra, S. K.: Generalized inverse of matrices and its applications. (1971) · Zbl 0236.15004
[12] Rao, C. R.; Mitra, S. K.; Bhimasankaram, P.: Determination of a matrix by its subclasses of generalized inverses. Sankhyā ser. A 34, 5-8 (1972) · Zbl 0274.15003
[13] Tian, Y.: Upper and lower bounds for ranks of matrix expressions using generalized inverses. Linear algebra appl. 355, 187-214 (2002) · Zbl 1016.15003
[14] Tian, Y.: More on maximal and minimal ranks of Schur complements with applications. Appl. math. Comput. 152, 675-692 (2004) · Zbl 1077.15005
[15] Y. Tian, The maximal and minimal ranks of a quadratic matrix expression with applications, submitted for publication.
[16] Tian, Y.; Cheng, S.: The maximal and minimal ranks of A - BXC with applications. New York J. Math. 9, 345-362 (2003) · Zbl 1036.15004