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.


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