Properties of the matrix A−XY*

AbstractThe main topic of this paper is the matrix V=A−XY*, where A is a nonsingular complex k×k matrix and X and Y are k×p complex matrices of full column rank. Because properties of the matrix V can be derived from those of the matrix Q=I−XY*, we will consider in particular the case where A=I. For the case that Y*X=I, so that Q is singular, we will derive the Moore–Penrose inverse of Q. The Moore–Penrose inverse of V in case Y*A−1X=I then easily follows. Finally, we will focus on the eigenvalues and eigenvectors of the real matrix D−xy′ with D diagonal