M-matrix

A Z-matrix $A$ is called an M-matrix if it satisfies any one ofthe 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 anon-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 suchthat $Av>0$ .
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 lowertriangular, $U$ is upper triangular, and the diagonal entries ofboth $L$ and $U$ are positive.
14.
The diagonal entries of $A$ are positive and $A D$ isstrictly diagonally dominant for some positive diagonal matrix $D$ .

Reference:

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

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

单词	Mmatrix
释义	M-matrix A Z-matrix $A$ is called an M-matrix if it satisfies any one ofthe 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 anon-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 suchthat $Av>0$ . 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 lowertriangular, $U$ is upper triangular, and the diagonal entries ofboth $L$ and $U$ are positive. 14. The diagonal entries of $A$ are positive and $A D$ isstrictly diagonally dominant for some positive diagonal matrix $D$ . Reference: M. Fiedler, Special Matrices and Their Applications inNumerical Mathematics, Martinus Nijhoff, Dordrecht, 1986. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis,Cambridge University Press, Cambridge, 1991.
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