%0 Journal Article
%T Decompositions of Some Special Block Tridiagonal Matrices
%A Hsin-Chu Chen
%J Advances in Linear Algebra & Matrix Theory
%P 54-65
%@ 2165-3348
%D 2021
%I Scientific Research Publishing
%R 10.4236/alamt.2021.112005
%X In this paper, we present a unified approach to decomposing a special class of
block tridiagonal matrices <i>K</i> (<i>¦Á</i> ,<i>¦Â</i> ) into block diagonal matrices using similarity
transformations. The matrices <i>K</i> (<i>¦Á</i> ,<i>¦Â</i> )¡Ê <i>R</i><sup><i>pq</i>¡Á <i>pq</i></sup> are of the form <i>K</i> (<i>¦Á</i> ,<i>¦Â</i> = block-tridiag[<i>¦Â B</i>,<i>A</i>,<i>¦Á B</i>] for three special pairs of (<i>¦Á</i> ,<i>¦Â</i> ): <i>K</i> (1,1), <i>K</i> (1,2) and <i>K</i> (2,2) , where the matrices <i>A</i> and <i>B</i>, <i>A</i>, <i>B</i>¡Ê <i>R</i><sup><i>p</i>¡Á <i>q</i></sup> , are general
square matrices. The decomposed block diagonal matrices <img src=\"Edit_00717830-3b3b-4856-8ecd-a9db983fef19.png\" width=\"15\" height=\"15\" alt=\"\" />(<i>¦Á</i> ,<i>¦Â</i> ) for
the three cases are all of the form: <img src=\"Edit_71ffcd27-6acc-4922-b5e2-f4be15b9b8dc.png\" width=\"15\" height=\"15\" alt=\"\" />(<i>¦Á</i> ,<i>¦Â</i> ) = <i>D</i><sub>1</sub> (<i>¦Á</i> ,<i>¦Â</i> ) ? <i>D</i><sub>2</sub> (<i>¦Á</i> ,<i>¦Â</i> ) ?---? <i>D</i><sub>q</sub> (<i>¦Á</i> ,<i>¦Â</i> ) ,
where <i>D<sub>k</sub></i> (<i>¦Á</i> ,<i>¦Â</i> ) = <i>A</i>+ 2cos ( <i>¦È<sub>k</sub></i> (<i>¦Á</i> ,<i>¦Â</i> )) <i>B</i>, in which <i>¦È<sub>k</sub></i> (<i>¦Á</i> ,<i>¦Â</i> ) , k = 1,2, --- q ,
depend on the values of <i>¦Á</i> and <i>¦Â</i>. Our decomposition method is closely related
to the classical fast Poisson solver using Fourier analysis. Unlike the fast
Poisson solver, our approach decomposes <i>K</i> (<i>¦Á</i> ,<i>¦Â</i> ) into <i>q</i> diagonal blocks,
instead of <i>p</i> blocks. Furthermore, our proposed approach does not require
matrices <i>A</i> and <i>B</i> to be symmetric and commute, and employs only the eigenvectors
of the tridiagonal matrix <i>T</i> (<i>¦Á</i> ,<i>¦Â</i> ) = tridiag[<i>¦Â b</i>, <i>a</i>,<i>¦Áb</i>] in a block form, where <i>a</i> and <i>b</i> 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.
%K Block Tridiagonal Matrices
%K Block Fourier Decomposition
%K Linear Systems
%K Eigenvalue Problems
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=109787