TITLE: Matrix exponential convergence
QUESTION [14 upvotes]: Help me please to prove that matrix exponential which defined as:
$e^{A}=\sum\limits_{k=0}^{\infty }\frac{A^{k}}{k!}$ converges for all matrix $A$
Thanks beforehand.

REPLY [2 votes]: HINT: Use one of the sub-multiplicative matrix norms to show that the series of real numbers in each position of $\sum_{k\ge 0}\frac{A^k}{k!}$ converges.