Quelltext der Seite Benutzer:K!r!93/sandbox/My first article

Eine '''''M''-Matrix''' ist in der [[Mathematik]], vor allem in der [[Lineare Algebra|Linearen Algebra]], ist eine [[Z-Matrix (Mathematik)|Z-Matrix]] bei der der [[Realteil]] aller ihrer [[Eigenwerte]] positiv ist.


== Literatur ==
* [[Gerd Fischer (Mathematiker)|Gerd Fischer]]: ''Lineare Algebra. (Eine Einführung für Studienanfänger).'' 13. durchgesehene Auflage. Vieweg, Braunschweig u. a. 2002, ISBN 3-528-97217-3.
* {{Literatur|Autor=Roger A. Horn, Charles R. Johnson|Titel=Matrix Analysis|Verlag=Cambridge University Press|Jahr=2012|ISBN=0-521-46713-6}}
* Hans-Rudolf Schwarz, Norbert Köckler: ''Numerische Mathematik.'' 5. überarbeitete Auflage. Teubner, Stuttgart u. a. 2004, ISBN 3-519-42960-8.

== Einzelnachweise ==
<references />

== Weblinks ==
* {{EoM|Autor=T. S. Pigolkina|Titel=Symmetric matrix|Url=http://www.encyclopediaofmath.org/index.php/Symmetric_matrix}}
* {{MathWorld|title=Symmetric matrix|id=SymmetricMatrix}}
* {{PlanetMath|title=Symmetric matrix|id=SymmetricMatrix|author=Daume}}

[[:Kategorie:Matrix]]

[[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 also 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> The name ''M''-matrix was seemingly originally chosen by [[Alexander Ostrowski]] in reference to [[Hermann Minkowski]], who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.<ref name="Berman">{{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>

== Characterizations ==
An M-matrix is commonly defined as follows:

'''Definition:''' Let ''A'' be a ''n × n'' real [[Z-matrix (mathematics)|Z-matrix]]. That is, ''A=(a<sub>ij</sub>)'' where ''a<sub>ij</sub> ≤ 0'' for all ''i ≠ j'', ''1 ≤ i,j ≤ n''. Then matrix ''A'' is also an ''M-matrix'' if it can be expressed in the form ''A = sI - B'', where ''B=(b<sub>ij</sub>)'' with ''b<sub>ij</sub> ≥ 0'', for all ''1 ≤ i,j ≤ n'', where ''s'' is greater than the maximum of the moduli of the eigenvalues of ''B'', and ''I'' is an identity matrix.

For the [[non-singularity]] of ''A'', according to [[Perron-Frobenius theorem]], it must be the case that ''s > ρ(B)''. Also, for non-singular M-matrix, the diagonal elements ''a<sub>ii</sub>'' of ''A'' must be positive. Here we will further characterize only the class of non-singular M-matrices. 

Many statements that are  equivalent to this definition of non-singular M-matrices are known, and any one of these statement can serve as a starting definition of a non-singular M-matrix.<ref>{{Citation |first=M |last=Fiedler |first2=V. |last2=Ptak |title=On matrices with non-positive off-diagonal elements and positive principal minors |journal=Czechoslovak Mathematical Journal |volume=12 |issue=3 |pages=382-400 |year=1962}}.</ref> For example, Plemmons lists 40 such equivalences.<ref>{{Citation |first=R.J. |last=Plemmons |title=M-Matrix Characterizations. I -- Nonsingular M-Matrices |journal=Linear Algebra and its Applications |volume=18 |issue=2 |pages=175-188 |year=1977 |doi=10.1016/0024-3795(77)90073-8}}.</ref> These characterizations has been categorized by Plemmons in terms of their relations to the properties of: (1) positivity of principal minors, (2) inverse-positivity and splittings, 
(3) stability, and (4) semipositivity and diagonal dominance. It makes sense to categorize the properties in this way because the statements within a particular group are related to each other  even when matrix ''A'' is an arbitrary matrix, and not necessarily a Z-matrix. Here we mention a few characterizations from each category.

==Equivalences==
Let ''A'' be a ''n × n'' real [[Z-matrix (mathematics)|Z-matrix]], then the following statements are equivalent to ''A'' being a [[non-singular]] M-matrix:

''Positivity of Principal Minors''
* All the [[minor (linear algebra)|principal minors]] of ''A'' are positive. That is, the determinant of each submatrix of ''A'' obtained by deleting a set, possibly empty, of corresponding rows and columns of ''A'' is positive.
* ''A + D'' is nonsingular for each nonnegative diagonal matrix ''D''.
* Every real eigenvalue of ''A'' is positive.
* All the leading principal minors of A are positive.
* There exist lower and upper triangular matrices ''L'' and ''U'' respectively, with positive diagonals, such that ''A'' = ''LU''.

''Inverse-Positivity and Splittings''
* ''A'' is ''inverse-positive''. That is, ''A<sup>−1</sup>'' exists and ''A<sup>−1</sup> ≥ 0''.
* ''A'' is ''monotone''. That is, ''Ax ≥ 0'' implies ''x ≥ 0''.
* ''A'' has a ''convergent regular splitting''. That is, ''A'' has a representation ''A = M - N'', where ''M<sup>−1</sup> ≥ 0, N ≥ 0'' with ''M<sup>−1</sup>N'' ''convergent''. That is, ''ρ(M<sup>−1</sup>N) < 1''.
* There exist inverse-positive matrices ''M<sub>1</sub>'' and ''M<sub>2</sub>'' with ''M<sub>1</sub> ≤ A ≤ M<sub>2</sub>''.
* Every regular splitting of ''A'' is convergent.

''Stability''
* There exists a positive diagonal matrix ''D'' such that ''AD + DA<sup>T</sup>'' is positive definite.
* ''A'' is ''positive stable''. That is, the real part of each eigenvalue of ''A'' is positive.
* There exists a symmetric [[positive definite matrix]] ''W'' such that ''AW + WA<sup>T</sup>'' is positive definite.
* ''A + I'' is non-singular, and ''G = (A + I)<sup>−1</sup>(A + I)'' is convergent.
* ''A + I'' is non-singular, and for ''G = (A + I)<sup>−1</sup>(A + I)'' there exists a positive definite symmetric matrix ''W'' such that ''W - G<sup>T</sup>WG'' is positive definite.

''Semipositivity and Diagonal Dominance''
* ''A'' is ''semi-positive''. That is, there exists ''x > 0'' with ''Ax > 0''.
* There exists ''x ≥ 0'' with ''Ax > 0''.
* There exists a positive diagonal matrix ''D'' such that ''AD'' has all positive row sums.
* ''A'' has all positive diagonal elements, and there exists a positive diagonal matrix ''D'' such that ''AD'' is ''strictly [[diagonally dominant]]''.
* ''A'' has all positive diagonal elements, and there exists a positive diagonal matrix ''D'' such that ''D<sup>−1</sup>AD'' is strictly diagonally dominant.

== Applications ==
The primary contributions to M-matrix theory has mainly come from mathematicians and economists. M-matrices are used in mathematics to establish bounds on eigenvalues and on the establishment of convergence criteria for [[iterative methods]] for the solution of large [[Sparse matrix|sparse]] [[systems of linear equations]]. M-matrices arise naturally in some discretizations of [[differential operators]], such as the [[Laplacian]], and as such are well-studied in scientific computing. M-matrices also occur in the study of solutions to [[linear complementarity problem]]. Linear complementarity problems arise in [[linear programming|linear]] and [[quadratic programming]], [[computational mechanics]], and in the problem of finding equilibrium point of a [[normal-form game|bimatrix game]]. Lastly, M-matrices occur in the study of finite [[Markov chains]] in the field of [[probability theory]] and [[operations research]] like [[queuing theory]]. Meanwhile, the economists have studied M-matrices in connection with gross substitutability, stability of a [[General equilibrium theory|general equilibrium]] and [[Input-output model| Leontief's input-output analysis]] in economic systems. The condition of positivity of all principal minors is also known as the [[Hawkins–Simon condition]] in economic literature.<ref>{{cite book |last=Nikaido |first=H. |title=Introduction to Sets and Mappings in Modern Economics |location=New York |publisher=Elsevier |year=1970 |isbn=0-444-10038-5 |pages=13–19 }}</ref> In engineering, M-matrices also occur in the problems of [[feedback control]] in [[control theory]] and is related to [[Hurwitz matrix]]. In [[computational biology]], M-matrices occur in the study of [[population dynamics]].

== See also ==
* If A is an M-matrix, then -A is a [[Metzler matrix]].
* A symmetric ''M''-matrix is sometimes called a [[Stieltjes matrix]].
* [[Hurwitz matrix]]
* [[Z-matrix (mathematics)|Z-matrix]]
* [[P-matrix]]
* [[Perron-Frobenius theorem]]