Reduced Symmetrizer Equation
symmetric matrix factorization, symmetrizer equation, symmetrizer, similarity symmetrization, diagonal matrix symmetrizer
Abstract
This paper revisits the real symmetrizer equation in the literature to transform it into a reduced symmetrizer equation. This reduction can be accomplished by decomposing the symmetrizer of the equation. The reduced equation has a diagonal matrix as its symmetrizer and can be further decomposed into more such equations. These reduced equations are coexisting and synchronized with the original symmetrizer equation. Associated results concerning the reduced symmetrizer equation are introduced. A numerical algorithm for symmetrizer computation is developed based on these results. Typical symmetrizer problems in the literature are solved using the algorithm and the results are presented.
Downloads
How to Cite
References
S Adhikari (2000) On Symmetrizable Systems of Second Kind. 67(4), 797-802.
R Bellman (1960) Introduction to Matrix Analysis.
E Desautels (1968) Symmetrizing matrices.
F Dopico, F Uhlig (2016) Computing matrix symmetrizers, part 2: New methods using eigendata and linear means; a comparison. 504, 590-622.
Ky Fan, A Hoffman (1955) Some Metric Inequalities in the Space of Matrices. 6(1), 111.
F Frobenius (1910) U ber die mit einer matrix vertauschbaren matrizen. 3, 1-15.
G Golub, C Loan (1985) Matrix Computations, 4th Edition.
James Howland (1961) A Method for Computing the Real Roots of Determinantal Equations. 68(3), 235-239.
J Howland, F Farrel (1963) 6. The Generalized Eigenvalue Problem. 233-263.
Jianguo Huang, Liwei Nong (2010) An iterative algorithm for solving a finite-dimensional linear operator equation http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll">T(x)=f with applications. 432(5), 1176-1188.
C Liu (2005) A Method of Symmetrization of Asymmetric Dynamical Systems. 12(4), 309-315.
F Ma, T Caughey (1995) Analysis of Linear Nonconservative Vibrations. 62(3), 685-691.
M Marcus (1960) Circular of the Bureau of Standards no. 24 4th edition:.
Marvin Marcus, N Khan (1960) On a commutator result of Taussky and Zassenhaus. 10(4), 1337-1346.
J Marty (1910) Existence de solutions singulieres pour certaines Equations de Fredholm. 150, 1031-1033.
J Neumann (1937) Some matrix-inequalities and metrization of matric-space. 1, 286-300.
Seymour Parter, J Youngs (1962) The symmetrization of matrices by diagonal matrices. 4(1), 102-110.
S Perlis (1952) Theory of Matrices.
H Pimbley (1959) Solution of an initial value problem for the multi-velocity neutron transport equation with a slab geometry. 8, 837-866.
R Nair (2020) Elementary uniform matrix symmetrization. 6, 140-156.
O Taussky (1972) The role of symmetric matrices in the study of general matrices. 5, 147-154.
O Taussky, H Zassenhaus (1959) On the similarity transformation between a matrix and its transpose. 9(3), 893-896.
S Sen, V Venkaiah (1988) On symmetrizing a matrix. 19(6), 554-561.
Frank Uhlig (2013) Computing matrix symmetrizers, finally possible via the Huang and Nong algorithm. 61(7), 954-969.
F Uhlig (2012) File S1: Matlab files and commands.
Published
2021-01-05
Issue
Section
License
Copyright (c) 2020 Authors and Global Journals Private Limited
This work is licensed under a Creative Commons Attribution 4.0 International License.