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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

[tex] WCW^T = C - I [/tex]

for the matrix C (an I is the identity matrix).

Background:The problem arised when I tried to find an expression for

[tex] C = \sum_{k=0}^{\infty} W^kW^{kT} [/tex]

in terms of the singular value decomposition of [tex]W = USV^T[/tex]. 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 [tex]C = QLQ.T[/tex] 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

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

Can you offer guidance or do you also need help?

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