On perturbations of non-diagonalizable stochastic matrices of order 3

We show that it is possible for every non-diagonalizable stochastic 3 × 3 matrix to
be perturbed into a diagonalizable stochastic matrix with the eigenvalues, arbitrarily close to the eigenvalues of the original matrix, with the same principal eigenspaces. An algorithm is presented to determine a perturbation matrix, which preserves these spectral properties. Additionally, a relation is proved between the eigenvectors and generalized eigenvectors of the original matrix and the perturbed matrix.