×

Matrices with multiple symmetry properties: applications of centro-Hermitian and per-Hermitian matrices. (English) Zbl 0957.15019

In this paper the authors introduce three matrix patterns, essentially asking that the matrix is either real or pure imaginary or zero, that combined with the twelve known symmetric patterns (symmetric, centrosymmetric, persymmetric, Hermitian, centro-Hermitian, per-Hermitian, skew-symmetric, skew-centrosymmetric, skew-persymmetric, skew-Hermitian, skew-centro-Hermitian, skew-per-Hermitian) form a Steiner triple system.
In addition, using a group of operators on the linear group over the complex numbers they reach types of matrices that satisfy sets of patterns, and that yield to unique decompositions into matrices of the same type. Also, they extend the symmetric patterns to vectors, and symmetric properties on matrices are deduced from the existence of basis of eigenvectors satisfying certain symmetric properties.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A45 Miscellaneous inequalities involving matrices
65F25 Orthogonalization in numerical linear algebra
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aitken, A. C., Determinants and Matrices (1939), Oliver and Boyd: Oliver and Boyd Edinburgh · Zbl 0022.10005
[2] Andrew, A. L., Eigenvectors of certain matrices, Linear Algebra Appl., 7, 151-162 (1973) · Zbl 0255.65021
[3] Barnett, S., Matrices, Methods and Applications (1990), Clarendon Press: Clarendon Press Oxford · Zbl 0706.15001
[4] Biedenham, L. C.; Louck, J. D., Angular momentum in quantum physics, (Theory and applications. Theory and applications, Encyclopedia of Physics and its Applications, vol. 8 (1981), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0474.00023
[5] Cantoni, A.; Butler, P., Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra Appl., 13, 275-288 (1976) · Zbl 0326.15007
[6] Collar, A. R., On centrosymmetric and centroskew matrices, Quart. J. Mech. Appl. Math., XV, 3, 264-281 (1962) · Zbl 0106.01205
[7] Cruse, A. B., Some combinatorial problems of centrosymmetric matrices, Linear Algebra Appl., 16, 65-77 (1977) · Zbl 0348.15011
[8] Datta, L.; Morgera, S. D., On the reducibility of centrosymmetric matrices - applications in engineering problems, (Circuits Systems Sig. Proc., 8 (1) (1989)), 71-96 · Zbl 0674.15005
[9] Gibson, W. M.; Pollard, B. R., Symmetry Principles in Elementary Particle Physics (1976), Cambridge University Press: Cambridge University Press Cambridge
[10] Good, I. J., The inverse of a centrosymmetric matrix, Technometrics, 12, 925-928 (1970) · Zbl 0194.05903
[11] Hill, R. D.; Bates, R. G.; Waters, S. R., On centrohermitian matrices, SIAM J. Matrix Anal. Appl., 11, 1, 128-133 (1990) · Zbl 0709.15021
[12] Hill, R. D.; Bates, R. G.; Waters, S. R., On perhermitian matrices, SIAM J. Matrix Anal. Appl., 11, 2, 173-179 (1990) · Zbl 0709.15022
[13] Hill, R. D.; Waters, S. R., On κ-real and κ-hermitian matrices, Linear Algebra Appl., 169, 17-29 (1992) · Zbl 0757.15011
[14] Horn, R. A.; Johnson, C. R., Matrix Analysis (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0704.15002
[15] Lander, E. S., Symmetric Designs: An Algebraic Approach, (London Math. Soc. Lecture Note Series, 74 (1983), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0502.05010
[16] Lee, A., Centrohermitian and skew-centrohermitian matrices, Linear Algebra Appl., 29, 205-210 (1980) · Zbl 0435.15019
[17] Ligh, S., Centro-symmetric matrices, Delta J. Sci., 3, 33-37 (1973) · Zbl 0303.15014
[18] Redei, L., (Algebra, vol. 1 (1967), Pergammon Press: Pergammon Press Oxford) · Zbl 0191.00502
[19] Stuart, J. L., Inflation matrices and ZME-matrices that commute with a permutation matrix, SIAM J. Matrix Anal. Appl., 9, 3, 408-418 (1988) · Zbl 0652.15009
[20] Weaver, J. R., Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, eigenvectors, MAA Monthly, 92, 711-717 (1985) · Zbl 0619.15021
[21] Weaver, J. R., Real eigenvalues of nonnegative matrices which commute with a symmetric matrix involution, Linear Algebra Appl., 110, 243-253 (1988) · Zbl 0661.15019
[22] Zehfuss, G., Zwei satze über determinanten, Z. Math. Phys., 7, 436-439 (1862)
[23] Zohar, S., Toeplitz matrix inversion: The algorithm of W.F. Trench, J. Assoc. Comput. Machinery, 16, 4, 592-601 (1969) · Zbl 0194.18102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.