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Article: Positive definite solutions of the nonlinear matrix equation X+AH X̄⁻¹ A=I
Title | Positive definite solutions of the nonlinear matrix equation X+AH X̄⁻¹ A=I |
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Authors | |
Keywords | Bound of solutions Complex matrix Nonlinear matrix equation Positive definite solutions |
Issue Date | 2013 |
Publisher | Elsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/amc |
Citation | Applied Mathematics and Computation, 2013, v. 219 n. 14, p. 7377-7391 How to Cite? |
Abstract | This paper is concerned with the positive definite solutions to the nonlinear matrix equation X+AHX̄-1A=I where X is the unknown and A is a given complex matrix. By introducing and studying a matrix operator on complex matrices, it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation W+BTW-1B=I which has been extensively studied in the literature, where B is a real matrix and is uniquely determined by A. It is also shown that if the considered nonlinear matrix equation has a positive definite solution, then it has the maximal and minimal solutions. Bounds of the positive definite solutions are also established in terms of matrix A. Finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed. |
Persistent Identifier | http://hdl.handle.net/10722/189206 |
ISSN | 2023 Impact Factor: 3.5 2023 SCImago Journal Rankings: 1.026 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Zhou, B | - |
dc.contributor.author | Cai, GB | - |
dc.contributor.author | Lam, J | - |
dc.date.accessioned | 2013-09-17T14:28:57Z | - |
dc.date.available | 2013-09-17T14:28:57Z | - |
dc.date.issued | 2013 | - |
dc.identifier.citation | Applied Mathematics and Computation, 2013, v. 219 n. 14, p. 7377-7391 | - |
dc.identifier.issn | 0096-3003 | - |
dc.identifier.uri | http://hdl.handle.net/10722/189206 | - |
dc.description.abstract | This paper is concerned with the positive definite solutions to the nonlinear matrix equation X+AHX̄-1A=I where X is the unknown and A is a given complex matrix. By introducing and studying a matrix operator on complex matrices, it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation W+BTW-1B=I which has been extensively studied in the literature, where B is a real matrix and is uniquely determined by A. It is also shown that if the considered nonlinear matrix equation has a positive definite solution, then it has the maximal and minimal solutions. Bounds of the positive definite solutions are also established in terms of matrix A. Finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed. | - |
dc.language | eng | - |
dc.publisher | Elsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/amc | - |
dc.relation.ispartof | Applied Mathematics and Computation | - |
dc.rights | © 2013. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ | - |
dc.subject | Bound of solutions | - |
dc.subject | Complex matrix | - |
dc.subject | Nonlinear matrix equation | - |
dc.subject | Positive definite solutions | - |
dc.title | Positive definite solutions of the nonlinear matrix equation X+AH X̄⁻¹ A=I | - |
dc.type | Article | - |
dc.identifier.email | Lam, J: james.lam@hku.hk | - |
dc.identifier.authority | Lam, J=rp00133 | - |
dc.description.nature | postprint | - |
dc.identifier.doi | 10.1016/j.amc.2013.01.021 | - |
dc.identifier.scopus | eid_2-s2.0-84875935680 | - |
dc.identifier.hkuros | 223488 | - |
dc.identifier.volume | 219 | - |
dc.identifier.issue | 14 | - |
dc.identifier.spage | 7377 | - |
dc.identifier.epage | 7391 | - |
dc.identifier.isi | WOS:000316668800010 | - |
dc.publisher.place | United States | - |
dc.identifier.issnl | 0096-3003 | - |