Matrix exponential

Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ be an $n \times n$ matrix over $\mathbb{K}$. The matrix exponential of $A$ is the $n \times n$ matrix $$e^A = \exp(A) = \sum_{k=0}^{\infty} \frac{1}{k!} A^k = I_n + A + \frac{1}{2} A^2 + \frac{1}{6} A^3 + \cdots,$$ with $A^0 = I_n$.

Remarks

The series converges for each matrix $A$, entry by entry. For a diagonal matrix $D$ with entries $d_i$, $e^D$ is diagonal with entries $e^{d_i}$. If $A = P D P^{-1}$, then $e^A = P e^D P^{-1}$. The function $x(t) = e^{tA} x_0$ solves the linear differential equation $x' = Ax$ with $x(0) = x_0$.

Explanations

L1 human

The matrix exponential uses the familiar series of $e^x$, with a matrix in place of the number $x$. It is the main tool for solving systems of linear differential equations. For $A = \begin{pmatrix} 0 & -t \\ t & 0 \end{pmatrix}$, $e^A$ is the rotation of the plane by the angle $t$.