When is a matrix diagonalizable?

Question

It is not clear for me. When is a matrix diagonalizable and when is it not ? I need some examples. (With 3x3 matrix and explanations please)

Answer 1

For example $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ is not diagonalizable.

$\begin{pmatrix}0&1\\1&0\end{pmatrix}$ is diagonalizable and in fact $\sim\begin{pmatrix}1&0\\0&-1\end{pmatrix}$. Note that the vectors $\begin{pmatrix}1\\\pm1\end{pmatrix}$ are eigenvectors of eigenvalue $\pm1$.

$\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ is diagonalizable over $\mathbb C$, but not over $\mathbb R$.

Answer 2

From Wikipedia, an $n \times n$ matrix $A$ over the field $F$ is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to $n$.

Answer 3

A matrix $M$ is diagonalisable if it has the form $M=P^{-1}DP$ with $P$ non-singular and $D$ a diagonal matrix.