The Pseudo-Inverse of a Product
Richard Bouldin · SIAM Journal on Applied Mathematics · 1973
Let A and B be bounded linear operators on a complex Hilbert space H, such that the range of each is a closed subspace of H. The following three conditions are necessary and sufficient for the pseudo-inverse of $AB$to be the pseudo-inverse of A followed by the pseudo-inverse of B : (i) the range of $AB$ must be closed; (ii) the range of $A^ * $ must be invariant under $BB^ * $; (iii) the intersection of the range of $A^ * $ and the kernel of $B^ * $ must be invariant under $A^ * A$. We use this basic result to obtain a simple technique for computing the pseudo-inverse of a given operator, particularly a given matrix.