M-matrix


A Z-matrix A is called an M-matrix if it satisfies any one of the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath conditions.

  1. 1.

    All principal minors of A are positive.

  2. 2.

    The leading principal minors of A are positive.

  3. 3.

    A can be written in the form A=kI-B, where B is a non-negative matrix whose spectral radius is strictly less than k.

  4. 4.

    All real eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A are positive.

  5. 5.

    The real part of any eigenvalue of A is positive.

  6. 6.

    A is non-singular and the inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of A is non-negative.

  7. 7.

    Av0 implies v0.

  8. 8.

    There exists a vector v with non-negative entries such that Av>0.

  9. 9.

    A+D is non-singular for every non-negative diagonal matrixMathworldPlanetmath D.

  10. 10.

    A+kI is non-singular for all k0.

  11. 11.

    For each nonzero vector v, vi(Av)i>0 for some i.

  12. 12.

    There is a positive diagonal matrix D such that the matrix DA+ATD is positive definitePlanetmathPlanetmath.

  13. 13.

    A can be factorized as LU, where L is lower triangular, U is upper triangular, and the diagonal entries of both L and U are positive.

  14. 14.

    The diagonal entries of A are positive and AD is strictly diagonally dominant for some positive diagonal matrix D.

Reference:

M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics, Martinus Nijhoff, Dordrecht, 1986.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.

Title M-matrix
Canonical name Mmatrix
Date of creation 2013-03-22 15:24:54
Last modified on 2013-03-22 15:24:54
Owner kshum (5987)
Last modified by kshum (5987)
Numerical id 7
Author kshum (5987)
Entry type Definition
Classification msc 15A57