M-matrix

A Z-matrix $A$ is called an M-matrix 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 $A=kI-B$ , where $B$ is a non-negative 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 non-singular and the inverse of $A$ is non-negative.
7.

$Av\geq 0$ implies $v\geq 0$ .
8.

There exists a vector $v$ with non-negative entries such that $Av>0$ .
9.

$A+D$ is non-singular for every non-negative diagonal matrix $D$ .
10.

$A+kI$ is non-singular for all $k\geq 0$ .
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 $L U$ , 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 $A D$ 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.