Minimal quasi-separable realizations for the inverse of a quasi-separable operator

We derive minimal quasi-separable (i.e. state space) representations for the upper and lower parts of the inverse of an invertible but otherwise general operator T which itself is given by its upper and lower minimal quasi-separable representations. We show that if the original representation is given in an adequate normal form, then the computation of the representation of the inverse can be done in a single downward or upward pass, involving only small, local computations. So called ‘intrinsic factors’ play an essential role in the derivation. We define them and show how they can be extracted. The results are given in closed form, provided one accepts the computation of a basis for a space and its orthogonal complement as numerically closed (QR type factorizations, common in ‘array computations’). The central workhorse is the classical square root algorithm utilized here in a generalized form.