If a square matrix $A$ of order $n$ has $r$ linearly independent eigenvectors, then it is possible to write $S^{+} A S=\Lambda$ where $S$ is the "eigenvector matrix of r linearly independent columns$^{\prime \prime}$ and $\Lambda$ the "eigenvalue matrix of the corresponding linearly independent eigenvectors " and $S^{+}$ is the left inverse of $S$ .
Is this a valid case of diagonalization? Does this diagonalization have any use?