Yu, Juan; Shen, Shu-Qian Solvability of systems of linear matrix equations subject to a matrix inequality. (English) Zbl 1357.15009 Linear Multilinear Algebra 64, No. 12, 2446-2462 (2016). The authors study the solvability conditions and the explicit expressions of the Hermitian solutions to the system of matrix equations \(AX=B\), \(XC=D\) subject to a matrix inequality \(MXM^* \geq N\) and the Hermitian nonnegative definite solutions to the system of matrix equations \(AX=B\), \(XC=D\) subject to a matrix inequality \(MXM^* \geq N\geq 0\). They make full use of the generalized inverse and the rank of matrices. As applications, some special cases of the above systems of matrix equations are considered. In addition, the maximal ranks and inertias of the Hermitian solutions are presented. Reviewer: Andreas Arvanitoyeorgos (Patras) MSC: 15A24 Matrix equations and identities 15B57 Hermitian, skew-Hermitian, and related matrices 15B48 Positive matrices and their generalizations; cones of matrices 15A03 Vector spaces, linear dependence, rank, lineability 15A45 Miscellaneous inequalities involving matrices Keywords:Hermitian matrix; Hermitian nonnegative definite matrix; rank; inertia; system of matrix equations; matrix inequality PDFBibTeX XMLCite \textit{J. Yu} and \textit{S.-Q. Shen}, Linear Multilinear Algebra 64, No. 12, 2446--2462 (2016; Zbl 1357.15009) Full Text: DOI References: [1] DOI: 10.1016/j.jmaa.2006.11.016 · Zbl 1120.47009 [2] DOI: 10.1137/0131050 · Zbl 0359.65033 [3] DOI: 10.1016/S0252-9602(08)60019-3 · Zbl 1150.15006 [4] DOI: 10.1016/0024-3795(84)90166-6 · Zbl 0543.15011 [5] DOI: 10.1016/j.amc.2006.12.046 · Zbl 1124.15009 [6] DOI: 10.1016/j.amc.2008.07.035 · Zbl 1157.65023 [7] DOI: 10.1016/j.camwa.2005.01.014 · Zbl 1138.15003 [8] DOI: 10.1016/j.amc.2011.08.018 · Zbl 1267.15017 [9] DOI: 10.1080/03081087.2012.663370 · Zbl 1264.15020 [10] DOI: 10.1155/2012/712651 · Zbl 1235.65070 [11] DOI: 10.1016/j.amc.2013.03.061 · Zbl 1290.15009 [12] DOI: 10.1016/j.laa.2008.01.030 · Zbl 1153.47012 [13] DOI: 10.1016/j.amc.2010.04.002 · Zbl 1193.65046 [14] Yu J, Math. Probl. Eng 2014 pp 1– (2014) [15] DOI: 10.1016/j.amc.2011.04.011 · Zbl 1222.15022 [16] DOI: 10.1016/j.laa.2010.01.015 · Zbl 1197.47031 [17] DOI: 10.13001/1081-3810.1492 · Zbl 1264.47022 [18] DOI: 10.1016/j.laa.2010.02.018 · Zbl 1205.15033 [19] DOI: 10.1016/j.laa.2010.12.010 · Zbl 1211.15022 [20] Tian Y-G, Math. Ineq. Appl 12 pp 537– (2012) [21] DOI: 10.1137/100808678 · Zbl 1252.65084 [22] Huang N, Abstract Appl. Anal 2014 pp 1– (2014) [23] DOI: 10.1137/080712945 · Zbl 1198.15010 [24] DOI: 10.1137/S0895479895270963 · Zbl 0912.93027 [25] DOI: 10.1016/S0024-3795(99)00108-1 · Zbl 0959.93032 [26] DOI: 10.1007/s10957-010-9760-8 · Zbl 1223.90077 [27] DOI: 10.1016/j.laa.2009.03.011 · Zbl 1180.15018 [28] DOI: 10.1080/03081087408817070 [29] DOI: 10.1137/S0895479802415545 · Zbl 1123.15001 [30] Tian Y-G, New York J. Math 9 pp 345– (2003) [31] DOI: 10.1016/S0024-3795(00)00033-1 · Zbl 0984.15011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.