Abstract
Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not preserved in general, we provide conditions on the underlying scalar function under which the property of interest will be preserved by the corresponding generalized matrix function.
References
[1] F. Andersson, M. Carlsson, and K.-M. Perfekt, Operator-Lipschitz estimates for the singular value functional calculus, Proc. Amer. Math. Soc., 144 (2016), pp. 1867-1875.10.1090/proc/12843Search in Google Scholar
[2] F. Arrigo and M. Benzi, Edge modiffication criteria for enhancing the communicability of digraphs, SIAM J.Matrix Anal. Appl., 37 (2016), pp. 443-468.10.1137/15M1034131Search in Google Scholar
[3] F. Arrigo, M. Benzi, and C. Fenu, Computation of generalizedmatrix functions, SIAM J.Matrix Anal. Appl., 37 (2016), pp. 836-860.10.1137/15M1049634Search in Google Scholar
[4] B. Arslan, V. Noferini, and F. Tisseur, The structured condition number of a differentiable map between matrix manifolds, with applications, MIMS EPrint 2017.36, Manchester Institute for Mathematical Sciences, 2017.Search in Google Scholar
[5] J. Aurentz, A. Austin, M. Benzi, and V. Kalantzis, Stable computation of generalized matrix functions via polynomial interpolation, Preprint, 2018.10.1137/18M1191786Search in Google Scholar
[6] P. J. Davis, Circulant Matrices, Wiley, New York, NY, 1979.Search in Google Scholar
[7] N. Del Buono, L. Lopez, and R. Peluso, Computation of the exponential of large sparse skew-symmetric matrices, SIAM J. Sci. Comput., 27 (2005), pp. 278-293.10.1137/030600758Search in Google Scholar
[8] N. Del Buono, L. Lopez and T. Politi, Computation of functions of Hamiltonian and skew-symmetric matrices, Math. Comp. Simul., 79 (2008), pp. 1284-1297.10.1016/j.matcom.2008.03.011Search in Google Scholar
[9] J. B. Hawkins and A. Ben-Israel, On generalized matrix functions, Linear and Multilinear Algebra, 1 (1973), pp. 163-171.10.1080/03081087308817015Search in Google Scholar
[10] N. J. Higham, Functions ofMatrices. Theory and Computation, Society for Industrial and AppliedMathematics, Philadelphia, PA, 2008.10.1137/1.9780898717778Search in Google Scholar
[11] N. J. Higham, D. S. Mackey, N. Mackey, and F. Tisseur, Functions preserving matrix groups and iterations for the matrix square root, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 849-877.10.1137/S0895479804442218Search in Google Scholar
[12] R. D. Hill, R. G. Bates, and S. R. Waters, On perhermitian matrices, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 173-179.10.1137/0611011Search in Google Scholar
[13] R. A. Horn and C. R. Johnson, Matrix Analysis. Second Edition, Cambridge University Press, 2013.Search in Google Scholar
[14] A. Lee, Centrohermitian and skew-centrohermitian matrices, Linear Algebra Appl., 29 (1980), pp. 205-210.10.1016/0024-3795(80)90241-4Search in Google Scholar
[15] V. Noferini, A formula for the Fréchet derivative of a generalized matrix function, SIAM J. Matrix Anal. Appl., 38 (2017), pp. 434-457.10.1137/16M1072851Search in Google Scholar
© by Michele Benzi, Ru Huang, published by De Gruyter
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