# Same eigenvalues = same jordan form ?

1. Oct 6, 2011

### JamesGoh

Say that for two mxm matrices, they have equivalent eigenvalues

If this is the case, is it safe to assume that the Jordan forms of both matrices will be the same ?

My reasoning comes from the fact that a general jordan block is represented
by the following matrix

λ 1 0 0
0 λ 1 0
0 0 λ 1
0 0 0 λ

and so on for an mxm matrix

2. Oct 7, 2011

### HallsofIvy

Staff Emeritus
No, it is not. The Jordan form depends upon the eigenvectors, not the eigenvalues. If a four by four matrix has $\lambda$ as its only eigenvalue then it can have Jordan forms of
$$\begin{bmatrix}\lambda & 0 & 0 & 0\\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda\end{bmatrix}$$
if it has four independent eigenvectors, or
$$\begin{bmatrix}\lambda & 1 & 0 & 0\\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda\end{bmatrix}$$
if it has three independent eigenvectors, or
$$\begin{bmatrix}\lambda & 1 & 0 & 0\\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda\end{bmatrix}$$
if it has two independent eigenvectors, or
$$\begin{bmatrix}\lambda & 1 & 0 & 0\\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & \lambda\end{bmatrix}$$
if it has only one independent eigenvector.