# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On solutions of the matrix equations $XF - AX = C$ and $XF - A\bar {X} =C$. (English) Zbl 1112.15018
The matrix equation $(*)\quad XF-AX=C$ is important in stability and control theory. When $F=A^*$, this is the Lyapunov matrix equation. The authors solve $(*)$ explicitly using the Kronecker map. The solution is expressed by a symmetric operator matrix, a controllability and an observability matrix. They express explicitly also the solution to the matrix equation $XF-A\bar{X}=C$ by means of the real representation of a complex matrix. The expression involves a symmetric operator matrix, two controllability and two observability matrices.

##### MSC:
 15A24 Matrix equations and identities 93B05 Controllability 93B07 Observability 93B25 Algebraic theory of control systems
Full Text:
##### References:
 [1] Bartels, R. H.; Stewart, G. W.: A solution of the equation AX+XB=C. Commun. ACM 15, 820-826 (1972) [2] Kleinman, D. L.; Rao, P. K.: Extensions to the bartels -- stewart algorithm for linear matrix equation. IEEE trans automat. Contr. 23, No. 1, 85-87 (1978) · Zbl 0369.65005 [3] Golub, G.; Nash, S.; Van Loan, C.: A Hessenberg -- Schur method for the problem AX+XB=C. IEEE trans. Automat. contr. 24, 909-913 (1979) · Zbl 0421.65022 [4] Sreeram, V.: Solution to Lyapunov equation with system matrix in companion form. IEE proc. D 138, 529-534 (1991) · Zbl 0754.34051 [5] Betser, A.; Cohen, N.; Zeheb, E.: On solving the Lyapunov and Stein equations for a companion matrix. Syst. cont. Lett. 25, 211-218 (1995) · Zbl 0877.93043 [6] Ma, E. C.: A finite series solution of the matrix equation AX - XB=C. SIAM J. Appl. math. 14, 490-495 (1966) · Zbl 0144.27003 [7] Zhou, B.; Duan, G. R.: An explicit solution to the matrix equation AX - XF=BY. Linear algebra appl. 402, 345-366 (2005) · Zbl 1076.15016 [8] Brockett, R. W.: Introduction to matrix analysis. (1970) [9] Young, N. J.: Formulae for the solution of Lyapunov matrix equation. Int. J. Contr. 31, 159-179 (1980) · Zbl 0432.93050 [10] Desouza, E.; Bhattacharyya, S.: Controllability and observability and the solution of AX - XB=C. Linear algebra appl. 39, 167-188 (1981) · Zbl 0468.15012 [11] Jones, J.; Charleslew, J. R.: Solutions of the Lyapunov matrix equation BX - XA=C. IEEE trans. Automat. contr. 27, No. 2, 464-466 (1982) [12] Hanzon, B.: A Faddeev sequence method for solving Lyapunov and Sylvester equations. Linear algebra appl., 401-430 (1996) · Zbl 0859.65038 [13] Jameson, A.: Solution of the equation AX - XB=C by inversion of an M$\times M$ or N$\times N$ matrix. SIAM J. Appl. math. 16, 1020-1023 (1968) · Zbl 0169.35202 [14] Jiang, T.; Wei, M.: On solutions of the matrix equations X - AXB=C and X-AX‾B=C. Linear algebra appl. 367, 225-233 (2003) · Zbl 1019.15002 [15] Hartwing, R. E.: Resultants and the solution of AX - XB=C. SIAM J. Appl. math. 23, No. 1, 105-117 (1972) [16] Huang, L.: The explicit solutions and solvability of linear matrix equations. Linear algebra appl. 311, 195-199 (2000) · Zbl 0958.15008 [17] Misra, P.; Quintana, E.; Van Dooren, P. M.: Numerically reliable computation of characteristic polynomials. Amer. contr. Conf. 6, 4025-4029 (1995)