- #1

TMO

- 45

- 1

## \begin{align}M =\begin{pmatrix} 2& -3& 0 \\ 3& -4& 0 \\ -2& 2& 1 \end{pmatrix} \end{align}. ##

Here is how I think the JCF is found.

**STEP 1: Find the characteristic polynomial**

It's ## \chi(\lambda) = (\lambda + 1)^3 ##

**STEP 2: Make an AMGM table and write an integer partition equation**

The AM is given by looking at the power. The GM is found by finding the nullspace for each eigenvalue. For this matrix this is the table:

Code:

```
+-----+------+------+
| λ | AM | GM |
+-----+------+------+
| -1 | 3 | 2 |
+-----+------+------+
```

which gives the integer partition equation ## J_{1, \lambda_{-1}} + J_{2, \lambda_{-1}} = 3 ##. Because there's only one integer partition possible (up to permutation: remember that the JCF is unique only up to permutation not in general), we can guess the JCF is

## \begin{align}J_M =\begin{pmatrix} -1& 1& 0 \\ 0& -1& 0 \\ 0& 0& -1 \end{pmatrix} \end{align}. ##

But I don't know how to compute the transition matrix. I know it involves generalized eigenvectors. Can someone help me?