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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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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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This is my definition for convergence. If $ \lbrace A^t \rbrace$ is a sequence of matrices and if $G$ is a matrix which has the same dimension as $A.$ We say that $ \lbrace A^t \rbrace$ convergs to $G$ when $ \lim_{ t \rightarrow \infty} A^t = G.$ –  Zizo Oct 6 '12 at 9:08
    
@did: It is not obvious to me. It seems we can only say $\|(A^T)^rCA^r\|$ approaches to zero when $r\rightarrow +\infty$. But it doesn't obviously imply the summation has a finite value. –  Shiyu Oct 6 '12 at 12:42
    
@Shiyu: You know that $\|C\|$ is finite and $\|A^T\|\cdot\|A\|\lt1$ but you are not sure that the series written in my post converges? This is odd... –  Did Oct 6 '12 at 14:15
    
@Sulaiman: Indeed. And my hint shows how to bound the norm of the difference between any partial sum and the limit. –  Did Oct 6 '12 at 14:16
    
@did: hi there. The part that confuses me is this: $(A^T)^r C A^r$ converges to zero when $r$ approaches infinity, but the summation or integral may not. Take a probably inappropriate example: $1/x$ approaches to zero when $x\rightarrow \infty$, but $\int_{1}^{\infty}1/x \mathrm{d}x$ is not finite. –  Shiyu Oct 6 '12 at 15:41

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