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On the covariance of the Moore-Penrose inverse. (English) Zbl 0547.15004

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

MSC:

15A09 Theory of matrix inversion and generalized inverses
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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
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