As we discussed, not every square matrix can be diagonalized, but every one can be put in Jordan canonical form

where $J$ is a block matrix of the form

where the $k^{\text{th}}$ Jordan block (of size $l_k\times l_k$) has the form

The special case, where all the Jordan blocks are $1\times 1$, is the case where $B$ is diagonalizable. When $B$ is diagonalizable, the matrix $S$ has $n$ columns, where the $k^{\text{th}}$ column is the eigenvector of $B$, corresponding to the eigenvalue ${\lambda }_k$:

or, equivalently,

Of course, this is a little bit ambiguous, as we can always multiply each $v_k$ by a non-zero constant. That ambiguity drops out of (1).

When a Jordan block has size $l_k>1$, we need, not one, but $l_k$ vectors to make up the corresponding columns of $S$. The first column is, again, given by the eigenvector

The next column is given by a vector, $w_{k,2}$, which satisfies

The column after that is given by $w_{k,3}$, which satisfies

and so on. Again, each of the $w_{k,i}$ is ambiguous by the addition of a multiple of $v_k$, but this drops out of (1).