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

  1. Sep 30, 2012 #1
    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
    [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,
  2. jcsd
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