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Convergent Matrix
In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation. Background When successive powers of a 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 regular splitting of a 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 \\ pt0 & \frac \end. \end Computing successive powers of T, we obtain :\begin & \mathbf^2 = \begin \frac & \frac \\ pt0 & \frac \end, \quad \mathbf^3 = \begin \frac & \frac \\ pt0 & \frac ... [...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 line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
