# Brainteaser: Solving an infinite sum of matrix products by consistency relation

1. Sep 30, 2012

### blue2script

Dear all,
I have been banging my head on this for a couple of hours with no result yet and given it is a very elegant problem (not a homework problem!) I would love to put it here as a brain teaser.

In a nutshell: Given a quadratic (and usually non-normal) matrix W, solve
$$WCW^T = C - I$$
for the matrix C (an I is the identity matrix).

Background: The problem arised when I tried to find an expression for
$$C = \sum_{k=0}^{\infty} W^kW^{kT}$$
in terms of the singular value decomposition of $$W = USV^T$$. Since the sum is infinite you immediately arrive at the consistency equation I put as a teaser. Since C is a real Hermitian matrix, it can be written as $$C = QLQ.T$$ where L is diagonal and Q a unitary matrix. The goal is to express Q and L in terms of U, S and V (where I take for granted that this gives the easiest representation which is to be found out).

Thanks for your help and happy brain teasing,
Wieland