On perturbations of non-diagonalizable stochastic matrices of order 3

Pieter-Jan Pauwelyn; M. A. Guerry

On perturbations of non-diagonalizable stochastic matrices of order 3

Pieter-Jan Pauwelyn, M. A. Guerry

Research output: Contribution to journal › Article › Scientific › peer-review

Abstract

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.

Original language	English
Article number	108633
Number of pages	6
Journal	Statistics & Probability Letters
Volume	157
Publication status	Published - Feb 2020
Externally published	Yes

Keywords

Stochastic matrices
non-diagonalizable matrices
perturbation theory
Markov chains

Cite this

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title = "On perturbations of non-diagonalizable stochastic matrices of order 3",

abstract = "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.",

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author = "Pieter-Jan Pauwelyn and Guerry, {M. A.}",

year = "2020",

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AU - Pauwelyn, Pieter-Jan

AU - Guerry, M. A.

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N2 - 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.

AB - 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.

KW - Stochastic matrices

KW - non-diagonalizable matrices

KW - perturbation theory

KW - Markov chains

M3 - Article

SN - 0167-7152

VL - 157

JO - Statistics & Probability Letters

JF - Statistics & Probability Letters

M1 - 108633

ER -

On perturbations of non-diagonalizable stochastic matrices of order 3

Abstract

Keywords

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