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A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. [2] In particular, a tridiagonal matrix is a direct sum of p 1-by-1 and q 2-by-2 matrices such that p + q/2 = n — the dimension of the tridiagonal.
In other situations, the system of equations may be block tridiagonal (see block matrix), with smaller submatrices arranged as the individual elements in the above matrix system (e.g., the 2D Poisson problem). Simplified forms of Gaussian elimination have been developed for these situations.
This method results in a very complicated set of equations in multiple dimensions, which are costly to solve. The advantage of the ADI method is that the equations that have to be solved in each step have a simpler structure and can be solved efficiently with the tridiagonal matrix algorithm. Consider the linear diffusion equation in two ...
A matrix that is both upper Hessenberg and lower Hessenberg is a tridiagonal matrix, of which the Jacobi matrix is an important example. This includes the symmetric or Hermitian Hessenberg matrices. A Hermitian matrix can be reduced to tri-diagonal real symmetric matrices. [7]
A band matrix with k 1 = k 2 = 0 is a diagonal matrix, with bandwidth 0. A band matrix with k 1 = k 2 = 1 is a tridiagonal matrix, with bandwidth 1. For k 1 = k 2 = 2 one has a pentadiagonal matrix and so on. Triangular matrices. For k 1 = 0, k 2 = n−1, one obtains the definition of an upper triangular matrix
The method only applies to matrices that can be represented as a (block) Toeplitz matrix. Such problems often arise in implicit solutions for partial differential equations on a lattice. For example fast solvers for Poisson's equation express the problem as solving a tridiagonal matrix, discretising the solution on a regular grid.
The method requires a transition rate matrix with tridiagonal block structure as follows = where each of B 00, B 01, B 10, A 0, A 1 and A 2 are matrices. To compute the stationary distribution π writing π Q = 0 the balance equations are considered for sub-vectors π i
It appears that the 2-d CN method is not going to lead to a tridiagonal system. Instead, we get a large square matrix, with small square matrices arranged tridiagonally on it: = with T a tridiagonal (5 X 5) matrix, I the (5 X 5) identity matrix and 0 the (5 X 5) matrix of zeros (this is obviously for the case of 5 points in each direction).