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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. Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
Block tridiagonal matrix: A block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements. Boolean matrix: A matrix whose entries are taken from a Boolean algebra. Cauchy matrix: A matrix whose elements are of the form 1/(x i + y j) for (x i), (y j) injective sequences (i.e., taking every value only once).
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; similarly, for k 1 = n−1, k 2 = 0 one obtains a lower triangular matrix. Upper and lower Hessenberg ...
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]
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The following matrix identity holds: ... The generalized continuant is precisely the determinant of the tridiagonal matrix ...
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