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Let $A$ be a matrix and let $C$ be a positive definite symmetric matrix. Then $B = \sum_{r=0}^ \infty (A^T)^rCA^r$ is symmetric and positive definite if $\|A\|<1.$

Any hints or proof? Thanks!

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up vote 4 down vote accepted

The norm condition guarantess convergence, hence $B$ is at least a matrix. Each summand and hence each partial sum is symmetric, hence so is $B$. Remains to show that $x^T B x> 0$ for all nonzero $x$. But $x^T B x= \sum_{r=0}^\infty (A^rx)^T C (A^rx)$ has all summands $\ge 0$ and the summand for $r=0$ is $>0$.

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