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.
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.
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?