Finding the Jordan canonical form of this upper triangular $3\times3$ matrix I am supposed to find the Jordan canonical form of a couple of matrices, but I was absent for a few lectures.
\begin{bmatrix} 
1 & 1 & 0 \\ 
0 & 1 & 2 \\
0 & 0 & 3
\end{bmatrix}
Since this is an upper triangular matrix, its eigenvalues are the diagonal entries. Hence $\lambda_{1,2}=1$ and $\lambda_3 = 3$, with corresponding eigenvectors
$(1,2,2)$ and $(1,0,0)$. Now what? I do not know how to proceed, nor what it means that my matrix is built up by Jordan blocks.
 A: The Jordan Canonical form of this matrix is 
$$
\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 3
\end{pmatrix}
.$$
You can read about Jordan Canonical forms here. Especially in section Complex matrices the properties listed there can help you find the Jordan Canonical form of most matrices.
A: Since this is a very small matrix there's actually no need to explicitly find the Jordan form by going through the routine tedious procedure. You know the characteristic polynomial is
$$p(x) = (x-1)^2(x-3)$$
and you can check that the minimal polynomial is the same. This means a few things.


*

*You have a single Jordan block corresponding to $3$. This is just $(3)$.

*You cannot have two Jordan blocks corresponding to $1$ since that would make the matrix diagonalizable (which it is not since the minimal polynomial does not split into distinct factors). Therefore you must have a single block of the form $\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$


These combined gives you the form (up to order of the blocks)
$$J=\begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0& 0 & 3\end{pmatrix}$$
In general, for small matrices like these (anything up to $6\times 6$) you can find the Jordan form through similar types of analysis using the facts


*

*The geometric multiplicity of an eigenvalue is the number of blocks corresponding to it.

*The algebraic multiplicity of an eigenvalue is the sum of the total sizes of the blocks.

*The exponent of the term corresponding to an eigenvalue in the minimal polynomial is the size of the largest block.

