Convergence of exponential of a Matrix

Question

Let $A$ be a square matrix. Is the exponential $e^A$ of $A$ is always converging?

Answer 1

We define $$ e^A = \sum_{k=0}^\infty \frac 1{k!}A^k $$ We wish to show that the sequence of partial sums $S_N = \sum_{k=0}^N \frac 1{k!}A^k$ converges to $S = e^A$. To that end, note that for the (sub-multiplicative) matrix norm $\|\cdot\|$ of your choosing, we have $$ \|S - S_N\| = \left\|\sum_{k=N+1}^\infty \frac 1{k!}A^k\right\| \leq \sum_{k=N+1}^\infty \frac 1{k!}\|A^k\| \leq \sum_{k=N+1}^\infty \frac 1{k!}\|A\|^k $$ The bound on the right approaches zero since the sum $\sum \frac 1{k!}\|A\|^k$ converges (to $e^{\|A\|}$, in fact).