MATH319 Slides 34 Proof of properties of exponential, concluded 36 Jordan blocks

35 Exponential of a diagonable matrix

Corollary

Suppose that $A$ has $n$ distinct eigenvalues $\lambda_{1},\dots,\lambda_{n}$ .

(i) Then there exists an invertible $n\times n$ matrix $S$ such that

\exp(tA)=S\begin{bmatrix}e^{t\lambda_{1}}&0&0&\dots\\ 0&e^{t\lambda_{2}}&0&\ddots\\ \vdots&\ddots&\ddots&\ddots\\ 0&\dots&0&e^{t\lambda_{n}}\end{bmatrix}S^{-1}

(ii) the entries of $\exp(tA)$ are complex linear combinations of $e^{t\lambda_{j}}$ for $j=1,\dots,n$ . Proof. There exists an invertible $n\times n$ matrix $S$ such that $A=SDS^{-1}$ where $D$ is the diagonal matrix with entries $\lambda_{1},\dots,\lambda_{n}$ . Hence $\exp(tA)=S\exp(tD)S^{-1}$ .