Angular factorization of matrices

Let Q denote a selfadjoint matrix (or operator of finite rank). The factorization of I + Q into the form $\text{I + Q = (I ? W) D(I ? W}{}^1\text{)}$ is considered. The well-known case where W is constrained to be lower triangular while D is diagonal is extended to quadrangular and other generalized-canonical forms. The case where Q is not selfadjoint is also developed. Applications for these results may be found in the realm of multivariate-stochastic approximation, and signal extraction.