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 \(\lambda_i\in \lambda(A)\)