The Exponential Series of a Real Square Matrix: Convergence, a Norm Bound and the Exponential Map

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$ and $M\in\mathcal{M}_{n}(\mathbb{R})$. For $k\in\mathbb{N}_{0}$, $M^{k}$ is the power of Powers of a Square Matrix, and $k!$ is the factorial of $k$, read in $\mathbb{R}$, where $k!>0$ by Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii); for $k\in\mathbb{N}$, $1\le k$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (ii), and $k-1\in\mathbb{N}_{0}$ is the difference. Partial sums, convergence and sums of series and absolute convergence are as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence, and $\exp$ is the exponential function. For $N\in\mathbb{N}$, $S_{N}(M)$ is the finite sum, for the addition of $\mathcal{M}_{n}(\mathbb{R})$ (the summand map exists by (i)),

$$S_{N}(M)=\sum_{k=1}^{N}\frac{1}{(k-1)!}\,M^{k-1},\qquad\text{also written}\qquad\sum_{k=0}^{N-1}\frac{M^{k}}{k!}.$$

(i) For every $N\in\mathbb{N}$, there is exactly one map $[N]\to\mathcal{M}_{n}(\mathbb{R})$ whose value at every $k\in[N]$ is $\frac{1}{(k-1)!}M^{k-1}$, the map whose finite sum is $S_{N}(M)$; and there is exactly one map $\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ whose value at every $X\in\mathcal{M}_{n}(\mathbb{R})$ is $S_{N}(X)$, that is, $S_{N}(M)$ for $X$ in place of $M$. There is exactly one sequence in $\mathcal{M}_{n}(\mathbb{R})$ whose $N$-th term is $S_{N}(M)$ for every $N\in\mathbb{N}$.

(ii) For all $i,j\in[n]$, there is exactly one sequence in $\mathbb{R}$ whose $k$-th term is $(M^{k-1})_{ij}/(k-1)!$ for every $k\in\mathbb{N}$, and its series

$$\sum_{k=1}^{\infty}\frac{(M^{k-1})_{ij}}{(k-1)!},\qquad\text{also written}\qquad\sum_{k=0}^{\infty}\frac{(M^{k})_{ij}}{k!},$$

converges absolutely, and in particular converges.

(iii) $S_{1}(M)=I_{n}$, and $S_{N+1}(M)=S_{N}(M)+\frac{1}{N!}M^{N}$ for every $N\in\mathbb{N}$.

(iv) There is exactly one $E\in\mathcal{M}_{n}(\mathbb{R})$ whose entry $E_{ij}$ is the sum of the series of (ii) for all $i,j\in[n]$, and the sequence $(S_{N}(M))_{N\in\mathbb{N}}$ converges to $E$ in $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$.

(v) The matrix $E$ of (iv) satisfies $\lVert E\rVert\le\exp(\lVert M\rVert)$.

(vi) There is exactly one map $\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ whose value at every $X\in\mathcal{M}_{n}(\mathbb{R})$ is the matrix $E$ of (iv) for $X$ in place of $M$.