A Numerical Method for the Inverse Stochastic Spectrum Problem

The inverse stochastic spectrum problem involves the construction of a stochastic matrix with a prescribed spectrum. The problem could be solved by first constructing a nonnegative matrix with the same prescribed spectrum. A differential equation aimed to bring forth the steepest descent flow in reducing the distance between isospectral matrices and nonnegative matrices, represented in terms of some general coordinates, is described. The flow is further characterized by an analytic singular value decomposition to maintain the numerical stability and to monitor the proximity to singularity. This flow approach can be used to design Markov chains with specified structure. Applications are demonstrated by numerical examples.

[1]  R. Rogers,et al.  An introduction to partial differential equations , 1993 .

[2]  George W. Soules.  Constructing symmetric nonnegative matrices , 1983 .

[3]  Raphael Loewy,et al.  A note on an inverse problem for nonnegative matrices , 1978 .

[4]  Moody T. Chu,et al.  Inverse Eigenvalue Problems , 1998, SIAM Rev..

[5]  D. E. Crabtree,et al.  On the characteristic roots of matrices , 1965 .

[6]  Shmuel Friedland,et al.  On an inverse problem for nonnegative and eventually nonnegative matrices , 1978 .

[7]  Mike Boyle,et al.  The spectra of nonnegative matrices via symbolic dynamics , 1991 .

[8]  A. Bunse-Gerstner,et al.  Numerical computation of an analytic singular value decomposition of a matrix valued function , 1991 .

[9]  M. Lewin.  On nonnegative matrices , 1971 .

[10]  Daniel Hershkowitz,et al.  Existence of matrices with prescribed eigenvalues and entries , 1983 .

[11]  F. Brauer,et al.  The Qualitative Theory of Ordinary Differential Equations: An Introduction , 1989 .

[12]  Charles R. Johnson,et al.  Possible spectra of totally positive matrices , 1984 .

[13]  K. Wright.  Differential equations for the analytic singular value decomposition of a matrix , 1992 .

[14]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[15]  Shmuel Friedland,et al.  On the eigenvalues of non-negative Jacobi matrices , 1979 .

[16]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[17]  G. Folland.  Introduction to Partial Differential Equations , 1976 .

[18]  M. Fiedler.  Eigenvalues of Nonnegative Symmetric Matrices , 1974 .

[19]  Volker Mehrmann,et al.  NUMERICAL METHODS FOR THE COMPUTATION OF ANALYTIC SINGULAR VALUE DECOMPOSITIONS , 1993 .

[20]  G. M. L. Gladwell,et al.  Inverse Problems in Vibration , 1986 .

[21]  Kenneth R. Driessel,et al.  Constructing symmetric nonnegative matrices with prescribed eigenvalues by differential equations , 1991 .