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# 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. 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 $LU$, 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.

## Mathematics Subject Classification

15A57*no label found*

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