# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 15A09 Matrix inversion, generalized inverses 15A03 Vector spaces, linear dependence, rank
Full Text:
##### 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