eigenvalues of stochastic matrix

Theorem: The spectrum of a stochastic matrixMathworldPlanetmath is contained in the unit disc in the complex planeMathworldPlanetmath.


Let A be a stochastic matrix and let m be an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A, with v eigenvectorMathworldPlanetmathPlanetmathPlanetmath; then, for any self-consistent matrix norm ., we have:


that is, since v is nonzero,


Now, for a (doubly) stochastic matrix,


whence the conclusionMathworldPlanetmath. ∎

Title eigenvalues of stochastic matrix
Canonical name EigenvaluesOfStochasticMatrix
Date of creation 2013-03-22 16:18:02
Last modified on 2013-03-22 16:18:02
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 7
Author Andrea Ambrosio (7332)
Entry type Theorem
Classification msc 60G99
Classification msc 15A51