Takagi Factorization of Matrices Depending on Parameters and Locating Degeneracies of Singular Values
Pages 1148 - 1161
Abstract
In this work, we consider the Takagi factorization of a matrix valued function depending on parameters. We give smoothness and genericity results and pay particular attention to the concerns caused by having either a singular value equal to 0 or multiple singular values. For these phenomena, we give theoretical results showing that their codimension is 2, and we further develop and test numerical methods to locate in parameter space values where these occurrences take place. Numerical study of the density of these occurrences is performed.
References
[1]
A. Bunse-Gerstner and W. Gragg, Singular value decompositions of complex symmetric matrices, J. Comput. Appl. Math., 21 (1988), pp. 41--54.
[2]
G. Cariolaro and G. Pierobon, Bloch-Messiah reduction of Gaussian unitaries by Takagi factorization, Phys. Rev. A, 94 (2016), 062109.
[3]
A.M. Chebotarev and A.E. Teretenkov, Singular value decomposition for the Takagi factorization of symmetric matrices, Appl. Math. Comput., 234 (2014), pp. 380--384.
[4]
R.C. Chen, Y.H. Kim, J.D. Lichtman, S.J. Miller, S. Sweitzer, and E. Winsor, Spectral statistics of non-Hermitian random matrix ensembles, Random Matrices Theory Appl., 08 (2019), 1950005.
[5]
J.L. Chern and L. Dieci, Smoothness and periodicity of some matrix decompositions, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 772--792.
[6]
L. Dieci and T. Eirola, On smooth decomposition of matrices, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 800--819.
[7]
L. Dieci, A. Papini, and A. Pugliese, Approximating coalescing points for eigenvalues of Hermitian matrices of three parameters, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 519--541.
[8]
L. Dieci, A. Papini, and A. Pugliese, Coalescing points for eigenvalues of banded matrices depending on parameters with application to banded random matrix functions, Numer. Algorithms, 80 (2019), pp. 1241--1266.
[9]
L. Dieci and A. Pugliese, Two-parameter SVD: Coalescing singular values and periodicity, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 375--403.
[10]
M. Hayes, Inhomogeneous plane waves, Arch. Ration. Mech. Anal., 85 (1984), pp. 41--79.
[11]
R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[12]
D.B. Horoshko, L. La Volpe, F. Arzani, N. Treps, C. Fabre, and M.I. Kolobov, Bloch-Messiah reduction for twin beams of light, Phys. Rev. A, 100 (2019), 013837.
[13]
V.A. Marchenko and L.A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mat. Sb. (N.S.), 72 (1967), pp. 507--536; Math. USSR-Sb., 1 (1967), pp. 457--483.
[14]
M.L. Mehta, Random Matrices, Elsevier Science, New York, 2004.
[15]
C.E. Reid and E. Brändas, On a theorem for complex symmetric matrices and its relevance in the study of decay phenomena, in Resonances, Lecture Notes in Physics, 325, Springer-Verlag, Berlin, 1989, pp. 475--483.
[16]
P.N. Walker and M.J. Sanchez, and M. Wilkinson, Singularities in the spectra of random matrices, J. Math. Phys., 37 (1996), pp. 5019--5032.
[17]
M. Wilkinson and E.J. Austin, Densities of degeneracies and near-degeneracies, Phys. Rev. A, 47 (1993), pp. 2601--2609.
[18]
W. Xu and S. Qiao, A divide-and-conquer method for the Takagi factorization, SIAM. J. Matrix Anal. Appl., 30 (2007), pp. 142--153.
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© 2022, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
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Published: 01 January 2022
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