M-matrix: Difference between revisions
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In [[mathematics]], especially [[linear algebra]], an '''''M''-matrix''' is a [[Z-matrix (mathematics)|''Z''-matrix]] with [[eigenvalue]]s whose [[real number|real]] parts are positive. ''M''-matrices are a subset of the class of [[P-matrix|''P''-matrices]], and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of [[Nonnegative matrix|positive matrices]]).<ref>{{Citation |first=Takao |last=Fujimoto |lastauthoramp=yes |first2=Ravindra |last2=Ranade |title=Two Characterizations of Inverse-Positive Matrices: The Hawkins-Simon Condition and the Le Chatelier-Braun Principle |journal=Electronic Journal of Linear Algebra |volume=11 |issue= |pages=59–65 |year=2004 |url=http://www.emis.ams.org/journals/ELA/ela-articles/articles/vol11_pp59-65.pdf }}.</ref> |
In [[mathematics]], especially [[linear algebra]], an '''''M''-matrix''' is a [[Z-matrix (mathematics)|''Z''-matrix]] with [[eigenvalue]]s whose [[real number|real]] parts are positive. ''M''-matrices are a subset of the class of [[P-matrix|''P''-matrices]], and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of [[Nonnegative matrix|positive matrices]]).<ref>{{Citation |first=Takao |last=Fujimoto |lastauthoramp=yes |first2=Ravindra |last2=Ranade |title=Two Characterizations of Inverse-Positive Matrices: The Hawkins-Simon Condition and the Le Chatelier-Braun Principle |journal=Electronic Journal of Linear Algebra |volume=11 |issue= |pages=59–65 |year=2004 |url=http://www.emis.ams.org/journals/ELA/ela-articles/articles/vol11_pp59-65.pdf }}.</ref> |
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Perhaps it should be mentioned: if a matrix is strictly diagonally dominant, and if the off-diagonal entries are negative or zero, then the matrix is an "M"-matrix. |
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A common characterization of an ''M''-matrix is a non-singular square matrix with non-positive off-diagonal entries and all principal minors positive, but many equivalences are known. The name ''M''-matrix was seemingly originally chosen by [[Alexander Ostrowski]] in reference to [[Hermann Minkowski]].<ref>{{Citation |first=Abraham |last=Bermon |lastauthoramp=yes |first2=Robert J. |last2=Plemmons |title=Nonnegative Matrices in the Mathematical Sciences |location=Philadelphia |publisher=Society for Industrial and Applied Mathematics |year=1994 |page=134,161 (Thm. 2.3 and Note 6.1 of chapter 6) |isbn=0-89871-321-8 }}.</ref> |
A common characterization of an ''M''-matrix is a non-singular square matrix with non-positive off-diagonal entries and all principal minors positive, but many equivalences are known. The name ''M''-matrix was seemingly originally chosen by [[Alexander Ostrowski]] in reference to [[Hermann Minkowski]].<ref>{{Citation |first=Abraham |last=Bermon |lastauthoramp=yes |first2=Robert J. |last2=Plemmons |title=Nonnegative Matrices in the Mathematical Sciences |location=Philadelphia |publisher=Society for Industrial and Applied Mathematics |year=1994 |page=134,161 (Thm. 2.3 and Note 6.1 of chapter 6) |isbn=0-89871-321-8 }}.</ref> |
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Revision as of 03:29, 1 August 2012
In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are positive. M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices).[1]
Perhaps it should be mentioned: if a matrix is strictly diagonally dominant, and if the off-diagonal entries are negative or zero, then the matrix is an "M"-matrix.
A common characterization of an M-matrix is a non-singular square matrix with non-positive off-diagonal entries and all principal minors positive, but many equivalences are known. The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski.[2]
A symmetric M-matrix is sometimes called a Stieltjes matrix.
M-matrices arise naturally in some discretizations of differential operators, particularly those with a minimum/maximum principle, such as the Laplacian, and as such are well-studied in scientific computing.
The LU factors of an M-matrix are guaranteed to exist and can be stably computed without the need for numerical pivoting and also have positive diagonal entries and non-positive off-diagonal entries. Furthermore, this holds even for incomplete LU factorization, where entries in the factors are discarded during factorization, providing useful preconditioners for an iterative solution.
See Also
References
- ^ Fujimoto, Takao; Ranade, Ravindra (2004), "Two Characterizations of Inverse-Positive Matrices: The Hawkins-Simon Condition and the Le Chatelier-Braun Principle" (PDF), Electronic Journal of Linear Algebra, 11: 59–65
{{citation}}: Unknown parameter|lastauthoramp=ignored (|name-list-style=suggested) (help). - ^ Bermon, Abraham; Plemmons, Robert J. (1994), Nonnegative Matrices in the Mathematical Sciences, Philadelphia: Society for Industrial and Applied Mathematics, p. 134,161 (Thm. 2.3 and Note 6.1 of chapter 6), ISBN 0-89871-321-8
{{citation}}: Unknown parameter|lastauthoramp=ignored (|name-list-style=suggested) (help).
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