The Exponential of a Real Square Matrix

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$ and $M\in\mathcal{M}_{n}(\mathbb{R})$. Powers $M^{k}$ are those of Powers of a Square Matrix; factorials, the series of the entries and the matrices $S_{N}(M)=\sum_{k=1}^{N}\frac{1}{(k-1)!}M^{k-1}$, $N\in\mathbb{N}$, are those of The Exponential Series of a Real Square Matrix: Convergence, a Norm Bound and the Exponential Map, and the sequence $(S_{N}(M))_{N\in\mathbb{N}}$ is that of The Exponential Series of a Real Square Matrix: Convergence, a Norm Bound and the Exponential Map (i). $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$ is a metric space by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence (i), and the limit of a convergent sequence in it is as in The Limit of a Convergent Sequence in a Metric Space (i).

(i) The exponential of $M$ is the matrix $e^{M}\in\mathcal{M}_{n}(\mathbb{R})$ that is the value at $M$ of the map of The Exponential Series of a Real Square Matrix: Convergence, a Norm Bound and the Exponential Map (vi). By The Exponential Series of a Real Square Matrix: Convergence, a Norm Bound and the Exponential Map (ii) and The Exponential Series of a Real Square Matrix: Convergence, a Norm Bound and the Exponential Map (iv), it is the only matrix whose entries are the sums of the series

$$(e^{M})_{ij}=\sum_{k=0}^{\infty}\frac{(M^{k})_{ij}}{k!}=\sum_{k=1}^{\infty}\frac{(M^{k-1})_{ij}}{(k-1)!}\qquad(i,j\in[n]).$$

(ii) Equivalently, $e^{M}$ is the limit of the sequence $(S_{N}(M))_{N\in\mathbb{N}}$ in $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$: by The Exponential Series of a Real Square Matrix: Convergence, a Norm Bound and the Exponential Map (iv), this sequence converges to $e^{M}$, so it is convergent, and its limit is $e^{M}$ by The Limit of a Convergent Sequence in a Metric Space (i):

$$e^{M}=\lim_{N\to\infty}\sum_{k=0}^{N-1}\frac{M^{k}}{k!}.$$