MATH319 Slides 38 Computing eigenvalues 40 Exponentials of complex matrices

39 Jordan Canonical Form

Theorem

Let $A$ be an $(n\times n)$ complex matrix. Then there exist $S$ an invertible $n\times n$ complex matrix, a partition of $n$ into $n=k_{1}+k_{2}+\dots+k_{r}$ and $k_{j}\times k_{j}$ Jordan blocks $J_{k_{j}}(\lambda_{j})$ where $\lambda_{j}$ is some eigenvalue of $A$ , such that $A$ is similar to the sum of Jordan blocks

A=S\begin{bmatrix}J_{k_{1}}(\lambda_{1})&0&\dots&0\\ 0&J_{k_{2}}(\lambda_{2})&0&\dots\\ \vdots&0&\ddots&0\\ 0&\dots&0&J_{k_{r}}(\lambda_{r})\end{bmatrix}S^{-1}.

See MATH220, Chapter 6.