Perturbations of non-diagonalizable stochastic matrices with preservation of spectral properties
Pieter-Jan Pauwelyn, Marie‐Anne Guerry · Linear and Multilinear Algebra · 2021
In Markov chain theory, stochastic matrices are used to describe inter-state transitions. Powers of such transition matrices are computed to determine the behaviour within a Markov system. For this, diagonalizable matrices are preferred because of their useful properties. The non-diagonalizable matrices are therefore undesirable. The aim is to determine a nearby diagonalizable matrix A~, starting from a non-diagonalizable matrix A. Previous studies tackled this problem, limited to 3×3 stochastic matrices. In this paper, these results are generalized for n×n stochastic matrices. Spectral properties of A are preserved in this process, such that A and A~ have coinciding semisimple eigenvalues and coinciding corresponding eigenvectors. This problem is examined and solved in this study and an algorithm is presented to find such a diagonalizable matrix A~.