Matrix Exponential

The matrix exponential of Square Matrix $X$ is defined by a power series

\begin{gather*} e^X\triangleq \sum_{n=0}^\infty\frac{X^n}{n!} \end{gather*}

Properties of this

\begin{gather*} \frac{dU}{dt}=XU(t)\quad U(0)=I \end{gather*}

diagonalizable Matrix → if $X$ is diagonalizable (which is guaranteed for, e.g., real symmetric or Hermitian matrices) then we can write $X=PDP^{-1}$ with $D=diag(\lambda_1, … \lambda_n)$ then

\begin{gather*} e^X=Pdiag(e^{\lambda_1},...,e^{\lambda_n})P^{-1} \end{gather*}

where $P$ is the basis matrix — a matrix whose columns are the eigenvectors $X$ w.r.t $\hat{A}$ .

Specifically, if $v_1,v_2,…,v_n$ are eigenvectors of $X$ corresponding to eigenvalues $\lambda_1,\lambda_2,…,\lambda_n$ then

\begin{gather*} P = \begin{bmatrix} | & | & & | \\ v_1 & v_2 & \cdots & v_n \\ | & | & & | \end{bmatrix} \end{gather*}

Hermitian and skew-Hermitian matrices → if $H=H^+$ then $e^{iH}$ is unitary this means

\begin{gather*} (e^{iH})^+e^{iH}=I \end{gather*}

Note that if $A$ is a matrix, then, via Taylor Series, we have

e^{At} = \sum_{k=0}^{\infty}\frac{A^k t^k}{k!} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots

\frac{d}{dt}e^{At} = \sum_{k=1}^{\infty}\frac{kA^k t^{k-1}}{k!} = A\sum_{k=1}^{\infty}\frac{A^{k-1}t^{k-1}}{(k-1)!} = Ae^{At} = e^{At}A