Abstract
Let A be a complex n × n matrix, and let A = B + iC, B = B*, C = C* be its Toeplitz decomposition. Then A is said to be (strictly) accretive if B > 0 and (strictly) dissipative if C > 0. We study the properties of matrices that satisfy both these conditions, in other words, of accretive-dissipative matrices. In many respects, these matrices behave as numbers in the first quadrant of the complex plane. Some other properties are natural extensions of the corresponding properties of Hermitian positive-definite matrices.
Similar content being viewed by others
REFERENCES
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1976; Russian translation: Mir, Moscow, 1972.
I. M. Glazman and Yu. I. Lyubich, Finite-Dimensional Linear Analysis [in Russian], Nauka, Moscow, 1969.
Yu. M. Arlinskii and A. B. Popov, “On sectorial matrices,” Linear Algebra Appl., 370 (2003), 133–146.
N. J. Higham, “Factorizing complex symmetric matrices with positive real and imaginary parts,” Math. Comp., 67 (1998), 1591–1599.
Kh. D. Ikramov and A. B. Kucherov, “Bounding the growth factor in Gaussian elimination for Buckley’s class of complex symmetric matrices,” Numer. Linear Algebra Appl., 7 (2000), 269–274.
A. George, Kh. D. Ikramov, and A. B. Kucherov, “On the growth factor in Gaussian elimination for generalized Higham matrices,” Numer. Linear Algebra Appl., 9 (2002), 107–114.
Kh. D. Ikramov and V. N. Chugunov, “Inequalities of Fisher and Hadamard types for accretive-dissipative matrices,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 384 (2002), no. 5, 585–586.
R. Mathias, “Matrices with positive-definite Hermitian part: Inequalities and linear systems,” SIAM J. Matrix Anal. Appl., 13 (1992), 640–654.
D. W. Masser and M. Neumann, “On the square roots of strictly quasiaccretive complex matrices,” Linear Algebra Appl., 28 (1979), 135–140.
C. R. Johnson, K. Okubo, and R. Reams, “Uniqueness of matrix square roots and an application,” Linear Algebra Appl., 323 (2001), 51–60.
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1994.
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, 1985; Russian translation; Mir, Moscow, 1990.
R. Bhatia and X. Zhan, “Compact operators whose real and imaginary parts are positive,” Proc. Amer. Math. Soc., 129 (2001), 2277–2281.
B. A. Mirman, “Numerical range and norm of a linear operator,” Voronezh. Gos. Univ. Trudy Sem. Funct. Anal., 10 (1968), 51–55.
R. Bhatia, Matrix Analysis, Springer-Verlag, Berlin, 1997.
Author information
Authors and Affiliations
Additional information
__________
Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 832–843.
Original Russian Text Copyright ©2005 by A. George, Kh. D. Ikramov.
Rights and permissions
About this article
Cite this article
George, A., Ikramov, K.D. On the Properties of Accretive-Dissipative Matrices. Math Notes 77, 767–776 (2005). https://doi.org/10.1007/s11006-005-0077-0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11006-005-0077-0