Generalize reflexive matrices are a special class of matrices ？that have the relation where？ and ？are some generalized reflection matrices. The nontrivial cases ( or ) of this class of matrices occur very often in many scientific and engineering applications. They are also a generalization of centrosymmetric matrices and reflexive matrices. The main purpose of this paper is to present block decomposition schemes for generalized reflexive matrices of various types and to obtain their decomposed explicit block-diagonal structures. The decompositions make use of unitary equivalence transformations and, therefore, preserve the singular values of the matrices. They lead to more efficient sequential computations and at the same time induce large-grain parallelism as a by-product, making themselves computationally attractive for large-scale applications. A numerical example is employed to show the usefulness of the developed explicit decompositions for decoupling linear least-square problems whose coefficient matrices are of this class into smaller and independent subproblems.
Weaver, J.R. (1985) Centrosymmetric (Cross-Symmetric) Matrices, Their Basic Properties, Eigenvalues, and Eigenvectors. The American Mathematical Monthly, 92, 711-717. https://doi.org/10.1080/00029890.1985.11971719
Weaver, J.R. (1988) Real Eigenvalues of Nonnegative Matrices Which Commute with a Symmetric Matrix Involution. Linear Algebra and Its Applications, 110, 243-253. https://doi.org/10.1016/0024-3795(83)90138-6
Tao, D. and Yasuda, M. (2002) A Spectral Characterization of Generalized Real Symmetric Centrosymmetric and Generalized Real Symmetric Skew-Centrosymmetric Matrices. SIAM Journal on Matrix Analysis and Applications, 23, 885-895. https://doi.org/10.1137/S0895479801386730
Chen, H.-C. and Sameh, A. (1989) A Domain Decomposition Method for 3D Elasticity Problems. In: Brebbia, C.A. and Peters, A., Eds., Applications of Supercomputers in Engineering: Fluid Flow and Stress Analysis Applications, Computational Mechanics Publications, Southampton University, Southampton, England, 171-188.