A Survey on M-Matrices

This is an expository-research paper which attempts to unify many of the more recent results on nonsingular and singular M-matrices (in the sense of Ostrowski and Schneider). Insight and direction are given to additional research on this very useful class of matrices and some new results are presented.

[1]  A. Ostrowski Über die determinanten mit überwiegender Hauptdiagonale , 1937 .

[2]  Lothar Collatz,et al.  Aufgaben monotoner Art , 1952 .

[3]  H. Schneider,et al.  An Inequality for Latent Roots Applied To Determinants with Dominant Principal Diagonal , 1953 .

[4]  R. Penrose A Generalized inverse for matrices , 1955 .

[5]  H. Schneider The Elementary Divisors, Associated with 0, of a Singular M-matrix , 1956, Proceedings of the Edinburgh Mathematical Society.

[6]  A. Householder On matrices with non-negative elements , 1958 .

[7]  K. Fan Topological proofs for certain theorems on matrices with non-negative elements , 1958 .

[8]  F. Gantmacher,et al.  Applications of the theory of matrices , 1960 .

[9]  A. Householder,et al.  A note concerning positive matrices and M-matrices , 1959 .

[10]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[11]  M. Fiedler,et al.  On matrices with non-positive off-diagonal elements and positive principal minors , 1962 .

[12]  D. Carlson A Note on M-Matrix Equations , 1963 .

[13]  On matrices with positive inverses , 1963 .

[14]  M. Lynn On the Schur product of H-matrices and non-negative matrices, and related inequalities , 1964, Mathematical Proceedings of the Cambridge Philosophical Society.

[15]  Vlastimil Pták,et al.  Some results on matrices of class $K$ and their application to the convergence rate of iteration procedures , 1966 .

[16]  D. E. Crabtree,et al.  CHARACTERISTIC ROOTS OF M-MATRICES , 1966 .

[17]  M. Fiedler,et al.  Diagonally dominant matrices , 1967 .

[18]  Olvi L. Mangasarian,et al.  Characterizations of Real Matrices of Monotone Kind , 1968 .

[19]  Shalhav Zohar,et al.  Toeplitz Matrix Inversion: The Algorithm of W. F. Trench , 1969, JACM.

[20]  I. J. Good,et al.  The Inverse of a Centrosymmetric Matrix , 1970 .

[21]  S. R. Searle Generalized Inverse Matrices , 1971 .

[22]  Patrick L. Odell,et al.  Generalized Inverse Matrices , 1971 .

[23]  T. Markham,et al.  Nonnegative matrices whose inverses are M-matrices , 1972 .

[24]  J. J. Moré Nonlinear generalizations of matrix diagonal dominance with application to Gauss-Seidel iterations. , 1972 .

[25]  T. O. Lewis,et al.  Weak Spectral Inverses , 1972 .

[26]  Luck J. Watford The Schur complement of a generalized M-matrix , 1972 .

[27]  Nancy M. Huang,et al.  Inversion of Persymmetric Matrices Having Toeplitz Inverses , 1972, JACM.

[28]  R. Plemmons Monotonicity and iterative approximations involving rectangular matrices , 1972 .

[29]  A. Berman,et al.  Monotonicity and the Generalized Inverse , 1972 .

[30]  J. Keilson,et al.  Markov chains and M-matrices: Inequalities and equalities , 1973 .

[31]  A Note on Pairs of Matrices and Matrices of Monotone Kind , 1973 .

[32]  K. S. Banerjee Generalized Inverse of Matrices and Its Applications , 1973 .

[33]  K. R. James Convergence of Matrix Iterations Subject to Diagonal Dominance , 1973 .