Mmatrix
A Zmatrix $A$ is called an Mmatrix if it satisfies any one of the following equivalent^{} conditions.

1.
All principal minors of $A$ are positive.

2.
The leading principal minors of $A$ are positive.

3.
$A$ can be written in the form $A=kIB$, where $B$ is a nonnegative matrix whose spectral radius is strictly less than $k$.

4.
All real eigenvalues^{} of $A$ are positive.

5.
The real part of any eigenvalue of $A$ is positive.

6.
$A$ is nonsingular and the inverse^{} of $A$ is nonnegative.

7.
$Av\ge 0$ implies $v\ge 0$.

8.
There exists a vector $v$ with nonnegative entries such that $Av>0$.

9.
$A+D$ is nonsingular for every nonnegative diagonal matrix^{} $D$.

10.
$A+kI$ is nonsingular for all $k\ge 0$.

11.
For each nonzero vector $v$, ${v}_{i}{(Av)}_{i}>0$ for some $i$.

12.
There is a positive diagonal matrix $D$ such that the matrix $DA+{A}^{T}D$ is positive definite^{}.

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.
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  Mmatrix 

Canonical name  Mmatrix 
Date of creation  20130322 15:24:54 
Last modified on  20130322 15:24:54 
Owner  kshum (5987) 
Last modified by  kshum (5987) 
Numerical id  7 
Author  kshum (5987) 
Entry type  Definition 
Classification  msc 15A57 