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A Z-matrix  is called an M-matrix if it satisfies any one of the following equivalent conditions.
- All principal minors of
 are positive.
- The leading principal minors of
 are positive.
 can be written in the form  , where  is a non-negative matrix whose spectral radius is strictly less than  .- All real eigenvalues of
 are positive.
- The real part of any eigenvalue of
 is positive.
 is non-singular and the inverse of  is non-negative. implies  .- There exists a vector
 with non-negative entries such that  .
 is non-singular for every non-negative diagonal matrix  . is non-singular for all  .- For each nonzero vector
 ,
 for some  .
- There is a positive diagonal matrix
 such that the matrix  is positive definite.
 can be factorized as  , where  is lower triangular,  is upper triangular, and the diagonal entries of both  and  are positive.- The diagonal entries of
 are positive and  is strictly diagonally dominant for some positive diagonal matrix  .
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.
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"M-matrix" is owned by kshum.
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(view preamble)
Cross-references: strictly diagonally dominant, diagonal, upper triangular, lower triangular, positive definite, diagonal matrix, vector, implies, inverse, non-singular, eigenvalue, real part, eigenvalues, real, strictly, spectral radius, matrix, positive, principal minors, equivalent, Z-matrix
There are 2 references to this entry.
This is version 4 of M-matrix, born on 2005-07-25, modified 2006-09-11.
Object id is 7257, canonical name is MMatrix.
Accessed 7179 times total.
Classification:
| AMS MSC: | 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices ) |
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Pending Errata and Addenda
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