On the covariance of the Moore-Penrose inverse (Q5896440)

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.