# 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)
Solutions and perturbation estimates for the matrix equations $X\pm A^*X^{-n}A=Q$. (English) Zbl 1063.15012
The authors consider the two matrix equations $X\pm A^*X^{-n}A=Q$ (where $A$ and $Q$ are complex $m\times m$-matrices) for $X\pm A^*X^{-n}A=Q$ existence of positive definite solutions. They obtain perturbation estimates for these solutions, as well as sufficient conditions for uniqueness of the positive definite solution of the “minus”-equation. The authors illustrate the results by numerical examples.

##### MSC:
 15A24 Matrix equations and identities
Full Text:
##### References:
 [1] Ferrante, A.; Levy, B.: Hermitian solutions of the equation X=Q+NX-1N\ast. Linear algebra appl. 247, 359-373 (1996) · Zbl 0876.15011 [2] El-Sayed, S. M.; Ran, A. C. M.: On an iteration method for solving a class of nonlinear matrix equations. SIAM J. Matrix anal. Appl. 23, 632-645 (2001) · Zbl 1002.65061 [3] Ivanov, I. G.; Hasanov, V. I.; Minchev, B. V.: On matrix equations $X{\pm}$A\astX-2A=I. Linear algebra appl. 326, 27-44 (2001) · Zbl 0979.15007 [4] V. Hasanov, I. Ivanov, F. Uhlig, Improved perturbation estimates for the matrix equations X{$\pm$}A\astX-1A=Q, Linear Algebra Appl., in press [5] V. Hasanov, I. Ivanov, Positive Definite Solutions of the Equation X+A\astX-nA=I, in: L. Vulkov, J. Wasniewski, P. Yalamov (Eds.), NAA 2000, LNCS 1988, Springer-Verlag, Berlin, Heidelberg, 2001, pp. 377--384 [6] Ran, A. C. M.; Reurings, M. C. B.: On the nonlinear matrix equation X+A$astF(X)$A=Q: solution and perturbation theory. Linear algebra appl. 346, 15-26 (2002)\$ · Zbl 1086.15013 [7] Xu, S. F.: Perturbation analysis of the maximal solution of the matrix equation X+A\astX-1A=P. Linear algebra appl. 336, 61-70 (2001)