Robinson, Donald W. On the covariance of the Moore-Penrose inverse. (English) Zbl 0547.15004 Linear Algebra Appl. 61, 91-99 (1984). Let A be an \(n\times n\) matrix and T an invertible matrix of the same order. If A is invertible, then, of course, \((TAT^{-1})^{-1}=TA^{- 1}T^{-1}\) and A is covariant under the general linear group. If A is arbitrary and the inverse is replaced by the Moore-Penrose inverse then A is no longer covariant under the full linear group. Given A the author finds the class of all nonsingular T under which A is covariant. Call this class C(A). Then C(A) contains nonzero scalar multiples of unitary matrices and is contained in the full linear group. The author finds necessary and sufficient conditions on A for these extreme bounds on C(A) to be obtained. Reviewer: G.P.Barker Cited in 1 ReviewCited in 5 Documents MSC: 15A09 Theory of matrix inversion and generalized inverses Keywords:invertible matrices; similarity; covariance; linear group; Moore-Penrose inverse; unitary matrices PDFBibTeX XMLCite \textit{D. W. Robinson}, Linear Algebra Appl. 61, 91--99 (1984; Zbl 0547.15004) Full Text: DOI References: [1] Ben-Israel, A.; Greville, T. N.E., Generalized Inverses: Theory and Applications (1974), Wiley: Wiley New York · Zbl 0305.15001 [2] Marcus, M.; Minc, H., A Survey of Matrix Theory and Matrix Inequalities (1964), Allyn and Bacon: Allyn and Bacon Boston · Zbl 0126.02404 [3] Marsaglia, G.; Styan, G., Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra, 2, 269-292 (1974) [4] Noble, B.; Daniel, J. W., Applied Linear Algebra (1977), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0413.15002 [5] Penrose, R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51, 406-413 (1955) · Zbl 0065.24603 [6] Schwerdtfeger, H., On the covariance of the Moore-Penrose inverse, Linear Algebra Appl., 52⧸53, 629-644 (1983) · Zbl 0525.15004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.