Let be a power series over with radius of convergence . Let be an matrix over , and let be a nonsingular matrix. Prove the following.
- If converges, then converges, and .
- If and converges, then and converge and .
- If is a diagonal matrix with diagonal entries , then converges, converges for each , and is diagonal with diagonal entries .
Suppose converges. Then (by definition) the sequence of matrices converges entrywise. Let , , and . Now . That is, the entry of is . Since each sequence converges, this sum converges as well. In particular, converges (again by definition). Now since for each , the corresponding sequences for each are equal for each term, and so have the same limit. Thus .
Now suppose . We have . Since converges in each entry, each of and converge in each entry. So and converge. Again, because for each the corresponding sequences and are the same, they converge to the same limit, and thus .
Finally, suppose is diagonal. Then in fact we have , and so by the previous part, . In particular, converges, and is diagonal with diagonal entries as desired.