ABSTRACT

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 $\tilde{A}$, starting from a non-diagonalizable matrix A. Previous studies tackled this problem, limited to $3\times 3$ stochastic matrices. In this paper, these results are generalized for $n\times n$ stochastic matrices. Spectral properties of A are preserved in this process, such that A and $\tilde{A}$ 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 $\tilde{A}$.

摘要

在马尔可夫链理论中，随机矩阵用于描述状态间转换。计算此类转移矩阵的幂以确定马尔可夫系统内的行为。为此，首选可对角化矩阵，因为它们具有有用的特性。因此，不可对角化矩阵是不受欢迎的。目的是确定附近的可对角化矩阵$\widetilde{\mathrm{一种}}$，从不可对角化的矩阵A开始。以前的研究解决了这个问题，仅限于$\mathrm{3个}\times \mathrm{3个}$随机矩阵。在本文中，这些结果被推广到$n\times n$随机矩阵。A的光谱特性在此过程中得以保留，因此A和$\widetilde{\mathrm{一种}}$具有重合的半单特征值和重合的相应特征向量。在这项研究中检查和解决了这个问题，并提出了一种算法来找到这样一个可对角化的矩阵$\widetilde{\mathrm{一种}}$.