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H-matrix (iterative Method)
In mathematics, an ''H''-matrix is a matrix (mathematics), matrix whose comparison matrix is an M-matrix. It is useful in iterative methods. Definition: Let be a complex matrix. Then comparison matrix ''M''(''A'') of complex matrix ''A'' is defined as where for all and for all . If ''M''(''A'') is a M-matrix, ''A'' is a ''H''-matrix. Invertible matrix, Invertible H-matrix guarantees convergence of Gauss–Seidel method, Gauss–Seidel iterative methods. See also * Hurwitz-stable matrix * P-matrix * Perron–Frobenius theorem * Z-matrix (mathematics), Z-matrix * L-matrix * M-matrix * Comparison matrix References [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Matrix
In linear algebra, let be a complex matrix. The comparison matrix of complex matrix ''A'' is defined as :\alpha_ = \begin -, a_, &\text i \neq j, \\ , a_, &\text i=j. \end See also * Hurwitz-stable matrix * P-matrix * Perron–Frobenius theorem * Z-matrix * L-matrix * M-matrix * H-matrix (iterative method) In mathematics, an ''H''-matrix is a matrix (mathematics), matrix whose comparison matrix is an M-matrix. It is useful in iterative methods. Definition: Let be a complex matrix. Then comparison matrix ''M''(''A'') of complex matrix ''A'' is def ... References Matrices {{matrix-stub ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the previous ones. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations A\mathbf=\mathbf by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. Howe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Seidel Method
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either strictly diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. A publication was not delivered before 1874 by Seidel. Description The Gauss–Seidel method is an iterative technique for solving a square system of linear equations with unknown : A\mathbf x = \mathbf b . It is defined by the iteration L_* \mathbf^ = \mathbf - U \mathbf^, where \mathbf^ is the -th approximation or iteration of \mathbf,\,\mathbf^ is the next or -t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurwitz-stable Matrix
In mathematics, a Hurwitz-stable matrix, or more commonly simply Hurwitz matrix, is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix. Such matrices play an important role in control theory. Definition A square matrix A is called a Hurwitz matrix if every eigenvalue of A has strictly negative real part, that is, :\operatorname[\lambda_i] < 0\, for each eigenvalue . is also called a stable matrix, because then the ordinary differential equation, differential equation : is stability theory, asymptotically stable, that is, as If is a (matrix-valued) transfer function, then is called Hurwitz if the pole (complex analysis), poles of all elements of have negative real part. Note that it is not necessary that for a specific argument [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-matrix
In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of -matrices, with every principal minor \geq 0. Spectra of -matrices By a theorem of Kellogg, the eigenvalues of - and P_0- matrices are bounded away from a wedge about the negative real axis as follows: :If \ are the eigenvalues of an -dimensional -matrix, where n>1, then ::, \arg(u_i), < \pi - \frac,\ i = 1,...,n :If , , are the eigenvalues of an -dimensional -matrix, then :: Remarks The class of nonsingular ''M''-matrices is a subset of the class of -matrices. More precisely, all matrices that are both -matrices and[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perron–Frobenius Theorem
In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Hawkins–Simon condition); to demography ( Leslie population age distribution model); to social networks ( DeGroot learning process); to Internet search engines (PageRank); and even to ranking of football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. Statement Let positive and non-negative respectively describe matrices with exclusively positive real numbers as elements and matrices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z-matrix (mathematics)
In mathematics, the class of ''Z''-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: :Z=(z_);\quad z_\leq 0, \quad i\neq j. Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term ''quasinegative'' matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made. The Jacobian of a competitive dynamical system is a ''Z''-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is ''J'', then (−''J'') is a ''Z''-matrix. Related classes are ''L''-matrices, ''M''-matrices, ''P''-matrices, ''Hurwitz'' matrices and ''Metzler'' matrices. ''L''-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a ''Z''-matrix is a ... [...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 st ...—the negation of any L-matrix is a Metzler matrix References Matrices {{matrix-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Matrix
In linear algebra, let be a complex matrix. The comparison matrix of complex matrix ''A'' is defined as :\alpha_ = \begin -, a_, &\text i \neq j, \\ , a_, &\text i=j. \end See also * Hurwitz-stable matrix * P-matrix * Perron–Frobenius theorem * Z-matrix * L-matrix * M-matrix * H-matrix (iterative method) In mathematics, an ''H''-matrix is a matrix (mathematics), matrix whose comparison matrix is an M-matrix. It is useful in iterative methods. Definition: Let be a complex matrix. Then comparison matrix ''M''(''A'') of complex matrix ''A'' is def ... References Matrices {{matrix-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
