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Let $A$ and $C$ two matrices where $\|A\|<1.$ I know that $ \lim_{k \rightarrow \infty} A^k = 0.$ I want to show that $B = \sum_{r=0}^ \infty (A^T)^rCA^r$ converges.

How to do this? Thank you.

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    related: [Positive definite of infinite sum of matrices](http://math.stackexchange.com/questions/207174/positive-definite-of-infinite-sum-of-matrices/207178)2012-10-06

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Hint: The matrix norm is subadditive and submultiplicative hence $\|B\|\leqslant\sum\limits_{r\geqslant0}\|(A^T)^rCA^r\|$ and, for each $r\geqslant0$, $\|(A^T)^rCA^r\|\leqslant\|C\|\cdot\|A^T\|^r\cdot\|A\|^r$. Can you take it from here?

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    @Shiyu Wonderful.2012-10-06