Matrix analysis
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Matrix analysis and applied linear algebra
Matrix analysis and applied linear algebra
Nonlinear Matrix Iterative Processes and Generalized Coefficients of Ergodicity
SIAM Journal on Matrix Analysis and Applications
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
On a Question Concerning Condition Numbers for Markov Chains
SIAM Journal on Matrix Analysis and Applications
The Local Coefficient of Ergodicity of a Nonnegative Matrix
SIAM Journal on Matrix Analysis and Applications
Jordan Canonical Form of the Google Matrix: A Potential Contribution to the PageRank Computation
SIAM Journal on Matrix Analysis and Applications
Introduction to probabilistic automata (Computer science and applied mathematics)
Introduction to probabilistic automata (Computer science and applied mathematics)
Convergence Analysis of a PageRank Updating Algorithm by Langville and Meyer
SIAM Journal on Matrix Analysis and Applications
PageRank Computation, with Special Attention to Dangling Nodes
SIAM Journal on Matrix Analysis and Applications
On optimal condition numbers for Markov chains
Numerische Mathematik
Ordinal Ranking for Google's PageRank
SIAM Journal on Matrix Analysis and Applications
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Ergodicity coefficients for stochastic matrices determine inclusion regions for subdominant eigenvalues; estimate the sensitivity of the stationary distribution to changes in the matrix; and bound the convergence rate of methods for computing the stationary distribution. We survey results for ergodicity coefficients that are defined by $p$-norms, for stochastic matrices as well as for general real or complex matrices. We express ergodicity coefficients in the one-, two-, and infinity-norms as norms of projected matrices, and we bound coefficients in any $p$-norm by norms of deflated matrices. We show that two-norm ergodicity coefficients of a matrix $A$ are closely related to the singular values of $A$. In particular, the singular values determine the extreme values of the coefficients. We show that ergodicity coefficients can determine inclusion regions for subdominant eigenvalues of complex matrices, and that the tightness of these regions depends on the departure of the matrix from normality. In the special case of normal matrices, two-norm ergodicity coefficients turn out to be Lehmann bounds.