An algorithm for constructing nonnegative matrices with prescribed real eigenvalues

Provided with the real spectrum, this paper presents a numerical procedure based on the induction principle to solve two types of inverse eigenvalue problems, one for nonnegative matrices and another for symmetric nonnegative matrices. As an immediate application, our approach can offer not only the sufficient condition to solve inverse eigenvalue problems for nonnegative or symmetric nonnegative matrices, but also a numerical way to solve inverse eigenvalue problems for stochastic matrices. Numerical examples are presented to show the capacity of our method.

[1]  G. Golub,et al.  Structured inverse eigenvalue problems , 2002, Acta Numerica.

[2]  A. M. Nazari,et al.  On the inverse eigenvalue problem for nonnegative matrices of order two to five , 2012 .

[3]  Hazel Perfect,et al.  Methods of constructing certain stochastic matrices. II , 1953 .

[4]  Ricardo L. Soto,et al.  Existence and construction of nonnegative matrices with prescribed spectrum , 2003 .

[5]  Thomas J. Laffey,et al.  Construction of nonnegative symmetric matrices with given spectrum , 2007 .

[6]  Moody T. Chu,et al.  A Numerical Method for the Inverse Stochastic Spectrum Problem , 1998, SIAM J. Matrix Anal. Appl..

[7]  T. Raghavan,et al.  Nonnegative Matrices and Applications , 1997 .

[8]  C. Marijuán,et al.  The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs , 2007 .

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

[10]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[11]  Haesun Park,et al.  Nonnegative matrix and tensor factorizations, least squares problems, and applications , 2011 .

[12]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorizations : An algorithmic perspective , 2014, IEEE Signal Processing Magazine.

[13]  Attalla Atia,et al.  Inverse eigenvalue problems : theory and algorithms. , 1998 .

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

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

[16]  R. Bruce Kellogg,et al.  Matrices similar to a positive or essentially positive matrix , 1971 .

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

[18]  Ricardo L. Soto,et al.  A Note on the Symmetric Nonnegative Inverse Eigenvalue Problem 1 , 2011 .

[19]  Helena Šmigoc,et al.  The inverse eigenvalue problem for nonnegative matrices , 2004 .

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

[21]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[22]  Moody T. Chu,et al.  On computing minimal realizable spectral radii of non-negative matrices , 2005, Numer. Linear Algebra Appl..

[23]  Abraham Berman.  Nonnegative matrices : Old problems, NewApplications , 2006 .

[24]  Delin Chu,et al.  Low rank update of singular values , 2006, Math. Comput..

[25]  Wuwen Guo,et al.  An inverse eigenvalue problem for nonnegative matrices , 1996 .

[26]  Peter Bloomfield,et al.  A flexible observed factor model with separate dynamics for the factor volatilities and their correlation matrix , 2011, 1106.3259.

[27]  Nicolas Gillis,et al.  Hierarchical Clustering of Hyperspectral Images Using Rank-Two Nonnegative Matrix Factorization , 2013, IEEE Transactions on Geoscience and Remote Sensing.

[28]  Alberto Borobia,et al.  On the nonnegative eigenvalue problem , 1995 .

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

[30]  C. Marijuán,et al.  A map of sufficient conditions for the real nonnegative inverse eigenvalue problem , 2007 .

[31]  S. Janson,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[32]  Robert Orsi.  Numerical Methods for Solving Inverse Eigenvalue Problems for Nonnegative Matrices , 2006, SIAM J. Matrix Anal. Appl..

[33]  Sivaram K. Narayan,et al.  The nonnegative inverse eigenvalue problem , 2004 .