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.
References
[1]
Chen, H.-C. (1998) Generalized Reflexive Matrices: Special Properties and Applications. SIAM Journal on Matrix Analysis and Applications, 19, 140-153. https://doi.org/10.1137/S0895479895288759
[2]
Zehfuss, G. (1862) Zwei Sätze über determinanten. Zeitschrift für Angewandte Mathematik und Physik, VII, 436-439.
[3]
Aitken, A.C. (1949) Determinants and Matrices. 6th Edition, Wiley-Interscience, New York.
[4]
Good, I.J. (1970) The Inverse of a Centrosymmetric Matrix. Technometrics, 12, 925-928. https://doi.org/10.1080/00401706.1970.10488743
Andrew, A.L. (1973) Eigenvectors of Certain Matrices. Linear Algebra and Its Applications, 7, 151-162. https://doi.org/10.1016/0024-3795(73)90049-9
[7]
Pye, W.C., Boullino, T.L. and Atchison, T.A. (1973) The Pseudoinverse of a Centrosymmetric Matrix. Linear Algebra and Its Applications, 6, 201-204. https://doi.org/10.1016/0024-3795(73)90020-7
[8]
Cantoni, A. and Butler, P. (1976) Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices. Linear Algebra and Its Applications, 13, 275-288. https://doi.org/10.1016/0024-3795(76)90101-4
[9]
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
[10]
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
[11]
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
[12]
Chen, H-C. and Sameh, A. (1989) A Matrix Decomposition Method for Orthotropic Elasticity Problems. SIAM Journal on Matrix Analysis and Applications, 10, 39-64. https://doi.org/10.1137/0610004
[13]
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.
[14]
Gibson, P.M. (1974) Simultaneous Diagonalization of Rectangular Complex Matrices. Linear Algebra and Its Applications, 9, 45-53. https://doi.org/10.1016/0024-3795(74)90025-1
[15]
Horn, R.A. and Johnson, C.A. (1985) Matrix Analysis. Cambridge University Press, New York. https://doi.org/10.1017/CBO9780511810817
[16]
Eckart, C. and Young, G. (1939) A Principal Axis Transformation for Non-Hermitian Matrices. Bulletin of the American Mathematical Society, 45, 118-121. https://doi.org/10.1090/S0002-9904-1939-06910-3
![\"\"](\"Edit_a4757d80-86b4-404f-99b1-ad8647b59c97.png\")
？that have the relation ![\"\"](\"Edit_cd6fc113-15df-47df-a80a-7ddced8e23f1.png\")
where？ ![\"\"](\"Edit_1932f493-b649-406e-a665-726c39059f03.png\")
and ![\"\"](\"Edit_07ce6a5e-336d-40c8-b785-65d5b91aae0e.png\")
？are some generalized reflection matrices. The nontrivial cases (![\"\"](\"Edit_d98061bd-615c-436f-8561-998dfe494c1d.png\")
or ![\"\"](\"Edit_0b6ff004-6f1c-4e68-9f75-c565cadbc1d1.png\")
) 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.
