Diagonalizability of a matrix

• A nilpotent matrix $A=\{A\in M_n |A^2=0\}$ is not diagonalizable. 
• Two matrices $A$ and $B$ are similar whenever these exists a nonsingular matrix $P$ such that $P^{-1} AP=B$
• A matrix can be diagonalized if it is similar to a diagonal matrix $D$, i.e. $P^{-1}AP=D$
• Or equivalently, $AP_{*,j}=\lambda_j P_{*,j}$
• $A$ is diagonalizable if and only if $A$ possesses a complete set of eigenvectors.
• Or equivalently, the geometric multiplicity of $\lambda_i$ is equal to the algebraic multiplicity of $\lambda_i$ for each $\lambda_i\in \lambda(A)$