On the solving matrix equations by using the spectral representation

Subject Areas : Linear and multilinear algebra; matrix theory

A. M. Nazari ¹ , S. Mollaghasemi ² , F. Bahmani ³

1 - Department of Mathematics, Faculty of Science, Arak university, Arak, PO. Box 38156-8-8349, Iran
2 - Department of Mathematics, Faculty of Science, Arak university, Arak, PO. Box 38156-8-8349, Iran
3 - Department of Mathematics, Faculty of Science, Arak university, Arak, PO. Box 38156-8-8349, Iran

Received: 2018-06-03 Accepted : 2018-12-10 Published : 2018-11-01

Keywords: Lyapanov equation, square matrix equations, spectral representation,

Abstract :

‎The purpose of this paper is to solve two types of Lyapunov equations and quadratic matrix equations by using the spectral representation‎. ‎We focus on solving Lyapunov equations $AX+XA^*=C$ and $AX+XA^{T}=-bb^{T}$ for $A‎, ‎X \in \mathbb{C}^{n \times n}$ and $b \in \mathbb{C} ^{n \times s}$ with $s < n$‎, ‎which $X$ is unknown matrix‎. ‎Also‎, ‎we suggest the new method for solving quadratic matrix equations $AX^{2}+BX+C=0$‎, ‎where $A‎, ‎B‎, ‎C‎, ‎X \in \mathbb{C}^{n \times n}$ and $X$ is unknown matrix with similar method‎.

References:

[1] A.C. Antoulas, D. C. Sorensen, Projection methods for balanced model reduction, Technical report, Rice university, Houston, 2001.

[2] R. Bartels, G. W. Stewart, Solution of the matrix equation AX + XB = C, Comm A.C.M. 15 (9) (1972), 820-826.

[3] P. Benner, T. Damm, Lyapunov equations, energy functions and model order reduction of bilinear and stochastic system, SIAM J. Control Optim. 49 (2011), 686-711.

[4] K. Glover, D. J. N. Limbeer, C. Doyle, E. M. Kasenally, M. G. Safonov, A characterisation of all solutions to the four block general distance problem, SIAM J. Control optim. 29 (1991), 283-324.

[5] R. A. Horn, D. R. Johnson, Topic in matrix analysis, Cambridge university press, cambridge, 1994.

[6] B. C. Moore, Principal component analysis in linear systems: controllabality, observability and model reduction, IEEE Trans. Auto. Control. 26 (1981), 17-31.

[7] A. Nazari, H. Fereydooni, M. Bayat, A manual approach for calculating the root of square matrix of dimension ≤ 3, Math. Sci. 2013, 2013:6 pages.

[8] S. Sambasiva Rao, M. Srinivas, V. Arvind Kumar, Square root of certain matrices, J. Global Res. Math. Arc. 2 (11) (2014), 19-24.

[9] B. Vandereycken, S. Vandewalle, A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations, SIAM J. Matrix Anal. Appl. 31 (5) (2010), 2553-2579.

[10] P. Van Dooren, Gramain based model reduction of large-scale dynamical systems, Numerical Analysis, Chapman and Hall, CRC press, London, 2000.