Why is $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ not diagonalizable I have a number of sufficient conditions as to when a matrix $A$ is diagonalizable, namely:


*

*When $A$ is symmetric

*When $A$ has distinct eigenvalues


Given $A = \begin{bmatrix} 0 & 1 \\ 0  & 0 \end{bmatrix}$
$A$ has nondistinct eigenvalues $\lambda = 0$ with algebraic multiplicity $2$, is there some conditions that says this is when the matrix fails to be diagonalizable?

You can check that $A^2 = 0$. If $A$ were diagonalizable, what would then be the options for values on the diagonal?
