## Compute the Jordan canonical form of a given matrix

Compute the Jordan canonical form of $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -2 \\ 0 & 1 & 3 \end{bmatrix}$.

We begin by computing the Smith normal form of $xI-A$. Evidently the following sequence of ERCOs achieves this.

1. $R_2 + xR_3 \rightarrow R_2$
2. $C_3 + (x-3)C_2 \rightarrow C_3$
3. $-C_2 \rightarrow C_2$
4. $R_2 \leftrightarrow R_3$
5. $R_1 \leftrightarrow R_2$
6. $C_1 \leftrightarrow C_2$

The resulting matrix is $\begin{bmatrix} 1 & 0 & 0 \\ 0 & x-1 & 0 \\ 0 & 0 & (x-1)(x-2) \end{bmatrix}$. So the invariant factors of $A$ are $x-1$ and $(x-1)(x-2)$, and the elementary divisors are thus $x-1$, $x-1$, and $x-2$. So the Jordan canonical form of $A$ is $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}$.

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