Matrix Exponential

Definition¹

Define the exponential function $\exp : M_{n \times n}(\mathbb{C}) \to M_{n \times n}(\mathbb{C})$ of a matrix as follows, and call it the matrix exponential (function).

$$ \exp (X) = e^{X} := \sum\limits_{m=0}^{\infty} \dfrac{X^{m}}{m!} \tag{1} $$

Here the above limit means the limit of matrices.

Explanation

The exponential function is defined by the power series form $e^{x} = \sum\limits_{n=0}^{\infty} \dfrac{x^{n}}{n!}$. It is defined this way because that makes it satisfy the defining property of the exponential function $\dfrac{d}{dt} e^{t} = e^{t}$.

$$ \dfrac{d}{dt} e^{Xt} = Xe^{Xt} $$

$X^{0}$ is defined as the identity matrix $I$. It follows the same properties as the exponential function defined on the real numbers. Note that $e^{X}$ itself is also a $n \times n$ matrix.

Properties

Assume $X, Y \in M_{n \times n}(\mathbb{C})$ and $\alpha, \beta \in \mathbb{C}$. Then the following matrix exponential 🔒(26/01/01)has the following properties.

(a) For all $X \in M_{n \times n}(\mathbb{C})$, the series $(1)$ converges and $e^{X}$ is continuous.

(b) $e^{O} = I$ ($O$ is the zero matrix)

(c) $(e^{X})^{\ast} = e^{X^{\ast}}$

(d) $e^{X}$ is an invertible matrix and $(e^{X})^{-1} = e^{-X}$.

(e) $e^{\alpha X + \beta Y} = e^{\alpha X} e^{\beta Y}$

(f) If $XY = YX$ then $e^{X+Y} = e^{X} e^{Y} = e^{Y} e^{X}$.

(g) If $C \in \operatorname{GL}(n, \mathbb{C})$ then $e^{C X C^{-1}} = C e^{X} C^{-1}$. Here $\operatorname{GL}(n, \mathbb{C})$ is the general linear group.

(h) For a diagonal matrix $D = [d_{ii}]$, $e^{D} = \begin{bmatrix} e^{d_{11}} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & e^{d_{nn}} \end{bmatrix}$.

Theorem

For $X \in M_{n \times n}(\mathbb{C})$, $e^{tX}$ is a smooth curve and the following holds.

$$ \dfrac{d}{dt} e^{tX} = X e^{tX} = e^{tX}X $$