If a matrix is Square Normal matrix then the matrix can be decomposed as:
$\hat{A}=\sum_{i=1}^{N}\lambda_i|\psi_i><\psi_i|$
where $\lambda_i$ are the eigenvalues of the matrix and $\psi_i$ corresponding eigenvectors.
Suppose the matrix A is not diagonalizable, but is still square. The Geometric multiplicity is strictly lesser than the Algebraic multiplicity.
What kind of decomposition can be obtained for such a matrix in terms of it's eigenvalues and eigenvectors?
(I am using the bra-ket notation of quantum mechanics in this question)