Definition:Matrix Exponential

Definition
Let $\mathbf A$ be a constant square matrix of order $n$.

The matrix exponential of $\mathbf A$, denoted $e^{t \mathbf A}$ or $e^{\mathbf A t}$, is defined as the unique solution to the initial value problem:
 * $(1): \quad \map {\dfrac \d {\d t} } X = \mathbf A X$
 * $(2): \quad \map X {\mathbf 0_n} = \mathbf I_n$

where:
 * $\mathbf I_n$ is the unit matrix of order $n$
 * $X$ is an order $n$ square matrix which is a function of the real variable $t$
 * $\mathbf 0_n$ is the zero matrix of order $n$
 * $\map {\dfrac \d {\d t} } X$ is the derivative of $X$ $t$.