Properties of the matrix A − XY*

The 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.