ALAMT  Vol.11 No.2 , June 2021
Decompositions of Some Special Block Tridiagonal Matrices
Abstract: 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 )∈ Rpq× 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, BRp× q , are general square matrices. The decomposed block diagonal matrices (α ,β ) for the three cases are all of the form: (α ,β ) = D1 (α ,β ) ⊕ D2 (α ,β ) ⊕---⊕ Dq (α ,β ) , where Dk (α ,β ) = A+ 2cos ( θk (α ,β )) B, in which θk (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.
Cite this paper: Chen, H. (2021) Decompositions of Some Special Block Tridiagonal Matrices. Advances in Linear Algebra & Matrix Theory, 11, 54-65. doi: 10.4236/alamt.2021.112005.

[1]   Buzbee, B.L., Golub, G.H. and Nielson, C.W. (1970) On Direct Methods for Solving Poisson’s Equations. SIAM Journal on Numerical Analysis, 7, 627-656.

[2]   Strang, G. (1986) Introduction to Applied Mathematics. Wellesley-Cambridge Press, Cambridge, MA.

[3]   Chen, H.C. (2002) A Block Fourier Decomposition Method. PARA 2002, 2367, 351-358.

[4]   Tolstov, G.P. and Series, F. (1962) Translated from the Russian by R.A. Silverman. Dover Publications, Inc., New York.

[5]   Marco, T. (2017) Properties of the Kronecker Product.

[6]   Meirovitch, L. (1980) Computational Methods in Structural Dynamics. Sijthoff & Noordhoff, Maryland.

[7]   Milne, W.E. (1970) Numerical Solution of Differential Equations. Dover Publications, Inc., New York.

[8]   Wait, R. and Mitchell, A.R. (1985) Finite Element Analysis and Applications. John Wiley & Sons, New York.

[9]   Chen, F. and Goshtasby, A. (1989) A Parallel B-Spline Surface Fitting Algorithm. ACM Transactions on Graphics, 8, 41-50.