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 the matrix equation $X+A\sp TX\sp{-1}A=I$. (English) Zbl 0863.15005
It is shown that the matrix equation $(*)$ $X+A^TX^{-1}A=I$ has a positive definite solution $(X>0)$ if and only if $A$ has a factorization $A=W^TZ$, where the matrix $W$ is nonsingular and the columns of $[W^T,Z^T]^T$ are orthonormal. In this case $X=W^TW$. It is also proved that equation $(*)$ has a solution $X>0$ if and only if there exist orthogonal matrices $P$ and $Q$ and diagonal matrices $\Gamma>0$, $\Sigma\ge 0$ with $\Gamma^2+\Sigma^2=I$ such that $A=P^T\Gamma Q\Sigma P$. Finally, it is shown that if $(*)$ has a solution $X>0$ then the following relations are valid $X-AA^T>0$, $I-AA^T-A^TA>0$, $r(A)\le 1/2$, $r(A+A^T)\le 1$, $r(A-A^T)\le 1$, where $r(A)$ is the spectral radius of $A$.

15A24Matrix equations and identities
Full Text: DOI
[1] Anderson, W. N.; Kleindorfer, G. B.; Kleindorfer, M. B.; Woodroofe, M. B.: Consistent estimates of the parameters of a linear system. Ann. math. Statist. 40, 2064-2075 (1969) · Zbl 0213.20703
[2] Anderson, W. N.; Morley, T. D.; Trapp, G. E.: Positive solutions to X = A - BX-1B\ast. Linear algebra appl. 134, 53-62 (1990) · Zbl 0702.15009
[3] Anderson, W. N.; Morley, T. D.; Trapp, G. E.: Ladder networks, fixed points, and the geometric mean. Circuits systems signal process 3, 259-268 (1983) · Zbl 0526.94017
[4] Ando, T.: Limit of cascade iteration of matrices. Numer. funct. Anal. optim. 21, 579-589 (1980)
[5] Ando, T.: Structure of operators with numerical radius one. Acta sci. Math. (Szeged) 34, 11-15 (1973) · Zbl 0258.47001
[6] Bucy, R. S.: A priori bound for the Riccati equation. Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, vol. III: probability theory, 645-656 (1972) · Zbl 0255.93025
[7] Engwerda, J. C.: On the existence of a positive definite solution of the matrix equation X = ATX-1A = I. Linear algebra appl. 194, 91-108 (1993) · Zbl 0798.15013
[8] Engwerda, J. C.; Ran, A. C. M.; Rijkeboer, A. L.: Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X + a\astx-1A = Q. Linear algebra appl. 186, 255-275 (1993) · Zbl 0778.15008
[9] Golub, G. H.; Van Loan, C. F.: Matrix computations. (1989) · Zbl 0733.65016
[10] Green, W. L.; Kamen, E.: Stabilization of linear systems over a commutative normed algebra with applications to spatially distributed parameter dependent systems. SIAM J. Control optim. 23, 1-18 (1985) · Zbl 0564.93054
[11] Horn, R. A.; Johnson, C. A.: Matrix analysis. (1985) · Zbl 0576.15001
[12] Ouellette, D. V.: Schur complements and statistics. Linear algebra appl. 36, 187-295 (1981) · Zbl 0455.15012
[13] Pusz, W.; Woronowitz, S. L.: Functional calculus for sequilinear forms and the purification map. Rep. math. Phys. 8, 159-170 (1975) · Zbl 0327.46032
[14] Rosenblum, M.; Rovnyak, J.: Hardy classes and operator theory. (1985) · Zbl 0586.47020
[15] Stewart, G. W.; Sun, J. G.: Matrix perturbation theory. (1990) · Zbl 0706.65013
[16] Trapp, G. E.: The ricatti equation and the geometric mean. Contemp. math. 47, 437-445 (1985)
[17] Zabezyk, J.: Remarks on the control of discrete time distributed parameter systems. SIAM J. Control 12, 721-735 (1974) · Zbl 0254.93027
[18] Zemanian, J.: Non-uniform semi-infinite grounded grids. SIAM J. Appl. math. 13, 770-788 (1982) · Zbl 0489.94029