|
In mathematics, especially linear algebra, an ''M''-matrix is a ''Z''-matrix with eigenvalues whose real parts are positive. ''M''-matrices are also 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).〔.〕 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.〔.〕 == Characterizations == An M-matrix is commonly defined as follows: Definition: Let ''A'' be a ''n × n'' real Z-matrix. That is, ''A=(aij)'' where ''aij ≤ 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=(bij)'' with ''bij ≥ 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 ''aii'' 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.〔.〕 For example, Plemmons lists 40 such equivalences.〔.〕 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「M-matrix」の詳細全文を読む スポンサード リンク
|