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Decompositions of Some Special Block Tridiagonal Matrices

机译：一些特殊块三角形矩阵的分解

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In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices K (α ,β ) into block diagonal matrices using similarity transformations. The matrices K (α ,β )∈ R ~(pq × pq ) are of the form K (α ,β = block-tridiag[β B ,A ,α B ] for three special pairs of (α ,β ): K (1,1), K (1,2) and K (2,2) , where the matrices A and B , A , B ∈ R ~(p × q ) , are general square matrices. The decomposed block diagonal matrices (α ,β ) for the three cases are all of the form: (α ,β ) = D _(1) (α ,β ) ⊕ D _(2) (α ,β ) ⊕---⊕ D _(q) (α ,β ) , where D_(k) (α ,β ) = A + 2cos ( θ_(k) (α ,β )) B , in which θ_(k) (α ,β ) , k = 1,2, --- q , depend on the values of α and β . Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis. Unlike the fast Poisson solver, our approach decomposes K (α ,β ) into q diagonal blocks, instead of p blocks. Furthermore, our proposed approach does not require matrices A and B to be symmetric and commute, and employs only the eigenvectors of the tridiagonal matrix T (α ,β ) = tridiag[β b , a ,αb ] in a block form, where a and b are scalars. The transformation matrices, their inverses, and the explicit form of the decomposed block diagonal matrices are derived in this paper. Numerical examples and experiments are also presented to demonstrate the validity and usefulness of the approach. Due to the decoupled nature of the decomposed matrices, this approach lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems using multiprocessors.
机译：在本文中，我们介绍了使用相似性转换将特殊类别的块三角形矩阵 k（α， r〜（ pq× pq）是形式的 k（α，β=嵌段嵌段 - 三个特殊对（α，βb，，β）= D _（1）（α，β）⊕ d _（2）（α， D _（q）（α，

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