A SHERMAN-MORRISON-WOODBURY IDENTITY FOR RANK AUGMENTING MATRICES WITH APPLICATION TO CENTERING

Matrices of the form A + (V1 + W1)G(V2 + W2)* are considered where A is a singular l x l matrix and G is a nonsingular k x k matrix, k less-than-or-equal-to l. Let the columns of V1 be in the column space of A and the columns of W1 be orthogonal to A. Similarly, let the columns of V2 be in the column space of A* and the columns of W2 be orthogonal to A*. An explicit expression for the inverse is given, provided that W(i)*W(i) has rank k. An application to centering covariance matrices about the mean is given.