Journal of Applied Mathematics

Volume 2014 (2014), Article ID 128249, 8 pages

http://dx.doi.org/10.1155/2014/128249

Research Article

## Notes on the Hermitian Positive Definite Solutions of a Matrix Equation

^{1}School of Mathematics and Statistics, Shandong University, Weihai 264209, China^{2}School of Mathematics, Shandong University, Jinan 250100, China

Received 15 January 2014; Revised 26 March 2014; Accepted 15 April 2014; Published 6 May 2014

Academic Editor: Alexander Timokha

Copyright © 2014 Jing Li and Yuhai Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- J. Zabczyk, “Remarks on the control of discrete-time distributed parameter systems,”
*SIAM Journal on Control and Optimization*, vol. 12, pp. 721–735, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. N. Anderson, Jr., G. B. Kleindorfer, P. R. Kleindorfer, and M. B. Woodroofe, “Consistent estimates of the parameters of a linear system,”
*Annals of Mathematical Statistics*, vol. 40, pp. 2064–2075, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. S. Bucy, “A priori bounds for the Riccati equation,” in
*Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability Theory*, pp. 645–656, Univ. California Press, Berkeley, Calif, USA, 1972. View at MathSciNet - D. V. Ouellette, “Schur complements and statistics,”
*Linear Algebra and Its Applications*, vol. 36, pp. 187–295, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. N. Anderson, Jr., T. D. Morley, and G. E. Trapp, “The cascade limit, the shorted operator and quadratic optimal control,” in
*Linear Circuits, Systems and Signal Processing: Theory and Application (Phoenix, AZ, 1987)*, C. I. Byrnes, C. F. Martin, and R. E. Saeks, Eds., pp. 3–7, North-Holland, Amsterdam, The Netherlands, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,”
*Reports on Mathematical Physics*, vol. 8, no. 2, pp. 159–170, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On direct methods for solving Poisson's equations,”
*SIAM Journal on Numerical Analysis*, vol. 7, pp. 627–656, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. A. Sakhnovich,
*Interpolation Theory and Its Applications*, vol. 428 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - X. Duan, Q. Wang, and A. Liao, “On the matrix equation arising in an interpolation problem,”
*Linear and Multilinear Algebra*, vol. 61, no. 9, pp. 1192–1205, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li, “Perturbation analysis of the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{{p}_{i}}{A}_{i}=Q$,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 979832, 11 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. C. Engwerda, A. C. M. Ran, and A. L. Rijkeboer, “Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation $X+{A}^{*}{X}^{-1}A=Q$,”
*Linear Algebra and Its Applications*, vol. 186, pp. 255–275, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. C. Engwerda, “On the existence of a positive definite solution of the matrix equation $X+{A}^{T}{X}^{-1}A=I$,”
*Linear Algebra and Its Applications*, vol. 194, pp. 91–108, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-H. Guo and P. Lancaster, “Iterative solution of two matrix equations,”
*Mathematics of Computation*, vol. 68, no. 228, pp. 1589–1603, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Ferrante and B. C. Levy, “Hermitian solutions of the equation $X=Q+{NX}^{-1}{N}^{*}$,”
*Linear Algebra and Its Applications*, vol. 247, pp. 359–373, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov and I. G. Ivanov, “On two perturbation estimates of the extreme solutions to the equations $X\pm {A}^{*}{X}^{-1}A=Q$,”
*Linear Algebra and Its Applications*, vol. 413, no. 1, pp. 81–92, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov, I. G. Ivanov, and F. Uhlig, “Improved perturbation estimates for the matrix equations $X\pm {A}^{*}{X}^{-1}A=Q$,”
*Linear Algebra and Its Applications*, vol. 379, pp. 113–135, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li and Y. H. Zhang, “The Hermitian positive definite solution and its perturbation analysis for the matrix equation $X-{A}^{*}{X}^{-1}A=Q$,”
*Mathematica Numerica Sinica*, vol. 30, no. 2, pp. 129–142, 2008 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Vaezzadeh, S. M. Vaezpour, R. Saadati, and C. Park, “The iterative methods for solving nonlinear matrix equation $X+{A}^{*}{X}^{-1}A+{B}^{*}{X}^{-1}B=Q$,”
*Advances in Difference Equations*, vol. 2013, article 229, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - I. G. Ivanov, V. I. Hasanov, and B. V. Minchev, “On matrix equations $X\pm {A}^{*}{X}^{-2}A=I$,”
*Linear Algebra and Its Applications*, vol. 326, no. 1-3, pp. 27–44, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-H. Zhang, “On Hermitian positive definite solutions of matrix equation $X-{A}^{*}{X}^{-2}A=I$,”
*Journal of Computational Mathematics*, vol. 23, no. 4, pp. 408–418, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-G. Liu and H. Gao, “On the positive definite solutions of the matrix equations ${X}^{s}\pm {A}^{T}{X}^{-t}A={I}_{n}$,”
*Linear Algebra and Its Applications*, vol. 368, pp. 83–97, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov, “Positive definite solutions of the matrix equations $X\pm {A}^{*}{X}^{-q}A=Q$,”
*Linear Algebra and Its Applications*, vol. 404, pp. 166–182, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li and Y. Zhang, “Perturbation analysis of the matrix equation $X-{A}^{*}{X}^{-p}A=Q$,”
*Linear Algebra and Its Applications*, vol. 431, no. 9, pp. 1489–1501, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Xiao and J. T. Wang, “On the matrix equation $X-{A}^{*}{X}^{-p}A=Q(p>1)$,”
*Chinese Journal of Engineering Mathematics*, vol. 26, no. 2, pp. 305–309, 2009. View at Google Scholar · View at MathSciNet - J. Li, “Solutions and improved perturbation analysis for the matrix equation $X-{A}^{*}{X}^{-p}A=Q(p>0)$,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 575964, 12 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Yin and L. Fang, “Perturbation analysis for the positive definite solution of the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{-1}{A}_{i}=Q$,”
*Journal of Applied Mathematics and Computing*, vol. 43, no. 1-2, pp. 199–211, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. Li and Y. Zhang, “Sensitivity analysis of the matrix equation from interpolation problems,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 518269, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. M. El-Sayed and A. C. M. Ran, “On an iteration method for solving a class of nonlinear matrix equations,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 23, no. 3, pp. 632–645, 2001/02. View at Publisher · View at Google Scholar · View at MathSciNet - A. C. M. Ran and M. C. B. Reurings, “On the nonlinear matrix equation $X+{A}^{*}\mathcal{F}\left(X\right)A=Q$: solutions and perturbation theory,”
*Linear Algebra and Its Applications*, vol. 346, pp. 15–26, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,”
*Proceedings of the American Mathematical Society*, vol. 132, no. 5, pp. 1435–1443, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. C. M. Ran, M. C. B. Reurings, and L. Rodman, “A perturbation analysis for nonlinear selfadjoint operator equations,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 28, no. 1, pp. 89–104, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. C. B. Reurings, “Contractive maps on normed linear spaces and their applications to nonlinear matrix equations,”
*Linear Algebra and Its Applications*, vol. 418, no. 1, pp. 292–311, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. C. B. Reurings,
*Symmetric matrix equations [Ph.D. thesis]*, VU University, Amsterdam, The Netherlands, 2003. - D. Zhou, G. Chen, G. Wu, and X. Zhang, “On the nonlinear matrix equation ${X}^{s}+{A}^{*}F\left(X\right)A=Q$ with $s\ge 1$,”
*Journal of Computational Mathematics*, vol. 31, no. 2, pp. 209–220, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{{\delta}_{i}}{A}_{i}=Q$,”
*Linear Algebra and Its Applications*, vol. 429, no. 1, pp. 110–121, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Lim, “Solving the nonlinear matrix equation $X=Q+{\sum}_{i=1}^{m}{M}_{i}{X}^{{\delta}_{i}}{M}_{i}^{*}$ via a contraction principle,”
*Linear Algebra and Its Applications*, vol. 430, no. 4, pp. 1380–1383, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Shi, F. Liu, H. Umoh, and F. Gibson, “Two kinds of nonlinear matrix equations and their corresponding matrix sequences,”
*Linear and Multilinear Algebra*, vol. 52, no. 1, pp. 1–15, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-F. Duan, Q.-W. Wang, and C.-M. Li, “Perturbation analysis for the positive definite solution of the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{{\delta}_{i}}{A}_{i}=Q$,”
*Journal of Applied Mathematics & Informatics*, vol. 30, no. 3-4, pp. 655–663, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Bhatia,
*Matrix Analysis*, vol. 169 of*Graduate Texts in Mathematics*, Springer, New York, NY, USA, 1997. View at Publisher · View at Google Scholar · View at MathSciNet