Suppose $A\_k>0$ (which means they are positive definitive square $n\times n$-matrices with $n>1$).
If $\sum\_{k=1}^\infty A\_k$ exists, then
$\sum\_{k=1}^\infty \|A\_k\| < +\infty$,
Where $\|A\|=\sup\_{\|x\|\leq 1}\langle Ax,x\rangle$.


Is this true? (I am not able to give any counterexample.)


Thank you!



When the sum of positive definite matrices converges, does the sum of the norm of the associate matrices converges?
You can bound $\|A\_k\| \leq C(n)\max\_{i,j} |(A\_k)\_{ij}|$ for some function of the dimension only $C(n)$, because all norms are equivalent in finite dimension. If I am not mistaken $C(n)=\sqrt{n}$, but it doesn't really matter here.


This maximum is attained on a (positive) diagonal entry, because of positive definiteness. 


Then you have $$\sum\_k \|A\_k\| \leq C(n) \sum\_k \max\_i (A\_k)\_{ii} \leq C(n)\sum\_k \sum\_i (A\_k)\_{ii} = C(n) \sum\_i (\sum\_k (A\_k)\_{ii}),$$ which is finite because $\sum\_k (A\_k)\_{ii}$ is finite for each $i$.


