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Inverse-positive Matrix
A real square matrix A is monotone (in the sense of Collatz) if for all real vectors v, Av \geq 0 implies v \geq 0, where \geq is the element-wise order on \mathbb^n. Properties A monotone matrix is nonsingular. ''Proof'': Let A be a monotone matrix and assume there exists x \ne 0 with Ax = 0. Then, by monotonicity, x \geq 0 and -x \geq 0, and hence x = 0. \square Let A be a real square matrix. A is monotone if and only if A^ \geq 0. ''Proof'': Suppose A is monotone. Denote by x the i-th column of A^. Then, Ax is the i-th standard basis vector, and hence x \geq 0 by monotonicity. For the reverse direction, suppose A admits an inverse such that A^ \geq 0. Then, if Ax \geq 0, x = A^ Ax \geq A^ 0 = 0, and hence A is monotone. \square Examples The matrix \left( \begin 1&-2\\ 0&1 \end \right) is monotone, with inverse \left( \begin 1&2\\ 0&1 \end \right). In fact, this matrix is an M-matrix (i.e., a monotone L-matrix). Note, however, that not all monotone matrices are M-matrices. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M-matrix
In mathematics, especially linear algebra, an ''M''-matrix is a ''Z''-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular ''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). 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 be a real Z-matrix. That is, where for all . Then matrix ''A'' is also an ''M-matrix'' if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix. For the non-singularity of , according to the Perron–Frobenius theorem, it must be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-matrix
In mathematics, the class of L-matrices are those matrices whose off-diagonal entries are less than or equal to zero and whose diagonal entries are positive; that is, an L-matrix ''L'' satisfies :L=(\ell_);\quad \ell_ > 0; \quad \ell_\leq 0, \quad i\neq j. See also * Z-matrix—every L-matrix is a Z-matrix * Metzler matrix In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): : \forall_\, x_ \geq 0. It is named after the American economist Lloyd Metzler. Metzler matrices appear in sta ...—the negation of any L-matrix is a Metzler matrix References Matrices {{matrix-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weakly Chained Diagonally Dominant Matrix
In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices. Definition Preliminaries We say row i of a complex matrix A = (a_) is strictly diagonally dominant (SDD) if , a_, >\textstyle, a_, . We say A is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with \geq instead. The directed graph associated with an m \times m complex matrix A = (a_) is given by the vertices \ and edges defined as follows: there exists an edge from i \rightarrow j if and only if a_ \neq 0. Definition A complex square matrix A is said to be weakly chained diagonally dominant (WCDD) if * A is WDD and * for each row i_1 that is ''not'' SDD, there exists a walk i_1 \rightarrow i_2 \rightarrow \cdots \rightarrow i_k in the directed graph of A ending at an SDD row i_k. Example The m \times m matrix :\begin1\\ -1 & 1\\ & -1 & 1\\ & & \ddots & \ddots\\ & & & -1 & 1 \end is WCD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
