zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 Hermitian positive definite solutions of matrix equation $X+A^{\ast} X^{-2} A=I$. (English) Zbl 1035.15017
The author considers the matrix equation $X+A^*X^{-2}A=I~(1)$ and its Hermitian positive definite solutions; here $A$ is an $n\times n$ complex matrix and $I$ is the identity matrix of order $n$. He shows that if $A$ is normal (i.e. $AA^*=A^*A$), then such a solution exists if and only if $\rho (A)\leq 2/3\sqrt{3}$ where $\rho (A)$ is the spectral radius of $A$. The author discusses in detail the basic fixed point iterations for the equation in the case when $A$ is nonnormal and $\Vert A\Vert \leq 2/3\sqrt{3}$ where $\Vert .\Vert $ stands for the spectral norm for square matrices (i.e. one has $0\leq A^*A\leq (4/27)I$). Some of the results of {\it I.G. Ivanov, V.I. Hasanov} and {\it B.V. Minchev} [ibid. 326, 27-44 (2001; Zbl 0979.15007)] are improved.

15A24Matrix equations and identities
65F10Iterative methods for linear systems
65F30Other matrix algorithms
Full Text: DOI
[1] Ivanov, I. G.; El-Sayed, S. M.: Properties of positive definite solution of the equation X+A*X-2A=I. Linear algebra appl. 279, 303-316 (1998) · Zbl 0935.65041
[2] Ivanov, I. G.; Hasanov, V. I.; Minchev, B. V.: On matrix equations $X{\pm}$A*X-2A=I. Linear algebra appl. 326, 27-44 (2001) · Zbl 0979.15007
[3] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045
[4] Hassanov, V.; Ivanov, I.: Positive definite solutions of the equation X+A*X-na=I. In: lect. Notes comput. Sci., numer. Anal. appl. 2000, 377-384 (2001) · Zbl 0978.65032