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Block Decompositions and Applications of Generalized Reflexive Matrices

DOI: 10.4236/alamt.2018.83011, PP. 122-133

Keywords: Generalized Reflexive Matrices, Reflexive Matrices, Centrosymmetric Matrices, Generalized Simultaneous Diagonalization, Simultaneous Diagonalization, Linear Least-Square Problems

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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.


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