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Convergent Matrix
In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under Matrix multiplication#Powers of a matrix, matrix exponentiation. Background When successive powers of a matrix (mathematics), matrix T become small (that is, when all of the entries of T approach zero, upon raising T to successive powers), the matrix T converges to the zero matrix. A matrix splitting, regular splitting of a invertible matrix, non-singular matrix A results in a convergent matrix T. A semi-convergent splitting of a matrix A results in a semi-convergent matrix T. A general iterative method converges for every initial vector if T is convergent, and under certain conditions if T is semi-convergent. Definition We call an ''n'' × ''n'' matrix T a convergent matrix if for each ''i'' = 1, 2, ..., ''n'' and ''j'' = 1, 2, ..., ''n''. Example Let :\begin & \mathbf = \begin \frac & \frac \\[4pt] 0 & \frac \end. \end Computing successive powers of T, we obtain :\begin & \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Matrices
This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called ''entries''. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include the identity matrix given by : I_n = \begin 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end. and the zero matrix of dimension m \times n. For example: : O_ = \begin 0 & 0 & 0 \\ 0 & 0 & 0 \end. Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the product of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including physics and chemistry, have particular matrices that are applied chiefly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrices
Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchise) * Matrix (mathematics), a rectangular array of numbers, symbols or expressions Matrix (or its plural form matrices) may also refer to: Science and mathematics * Matrix (mathematics), algebraic structure, extension of vector into 2 dimensions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryotic organism's cells * Matrix (chemical analysis), the non-analyte components of a sample * Matrix (geology), the fine-grained material in which larger objects are embedded * Matrix (composite), the constituent of a composite material * Hair matrix, produces hair * Nail matrix, part of the nail in anatomy Arts and entertainment Fictional entities * Matrix (comics), two comic book ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
