He, Zhuo-Heng; Wang, Qing-Wen A real quaternion matrix equation with applications. (English) Zbl 1317.15016 Linear Multilinear Algebra 61, No. 6, 725-740 (2013). Let \(\mathbb{H}^{m\times n}\) be the set of all \(m\times n\) matrices over the real quaternion algebra \[ \mathbb{H}=\{a_0+a_1i+a_2j+a_3k|\,i^2=j^2=k^2=ijk=-1,\, a_0,a_1,a_2,a_3\in\mathbb{R}\}. \] For \(A\in\mathbb{H}^{m\times n}\), it is denoted that \(A^{\eta}=-\eta A\eta\), and \(A^{\eta^*}=-\eta A^*\eta\), where \(\eta\in\{i,j,k\}\), and \(A^*\) is the conjugate transpose of \(A\), and the map \(A\mapsto A^{\eta^*}\) is an involution. A matrix \(A\in\mathbb{H}^{n\times n}\) is called \(\eta\)-Hermitian if \(A^{\eta^*}=A\) for \(\eta\in\{i,j,k\}\).In the paper, the real quaternion matrix equation \[ A_1X+(A_1X)^{\eta^*}+B_1YB_1^{\eta^*}+C_1ZC_1^{\eta^*}=D_1 \] is considered. For the case when \(D_1\) is \(\eta\)-Hermitian, necessary and sufficient conditions on matrices \(A_1\), \(B_1\), \(C_1\), and \(D_1\) are established for the equation above to be solvable with respect to the triplet \((X,Y,Z)\), where \(Y\) and \(Z\) are required to be \(\eta\)-Hermitian. The explicit solution is presented and the minimal ranks of the solutions \(Y\) and \(Z\) are found. Reviewer: Mikhail Tyaglov (Shanghai) Cited in 1 ReviewCited in 54 Documents MSC: 15A24 Matrix equations and identities 15A09 Theory of matrix inversion and generalized inverses 15B57 Hermitian, skew-Hermitian, and related matrices 11R52 Quaternion and other division algebras: arithmetic, zeta functions 15B33 Matrices over special rings (quaternions, finite fields, etc.) Keywords:linear matrix equation; quaternion matrix; \(\eta\)-Hermitian solution; Moore-Penrose inverse; ranks; real quaternion matrix equation PDFBibTeX XMLCite \textit{Z.-H. He} and \textit{Q.-W. Wang}, Linear Multilinear Algebra 61, No. 6, 725--740 (2013; Zbl 1317.15016) Full Text: DOI References: [1] DOI: 10.1016/j.sigpro.2004.04.001 · Zbl 1154.94331 · doi:10.1016/j.sigpro.2004.04.001 [2] DOI: 10.1109/70.127239 · doi:10.1109/70.127239 [3] DOI: 10.1137/S0895479895270963 · Zbl 0912.93027 · doi:10.1137/S0895479895270963 [4] Chu DL, SIAM J. Matrix Anal. Appl. 3 pp 1187– (2009) [5] DOI: 10.1016/S0024-3795(99)00108-1 · Zbl 0959.93032 · doi:10.1016/S0024-3795(99)00108-1 [6] DOI: 10.1088/0305-4470/33/15/306 · Zbl 0954.81008 · doi:10.1088/0305-4470/33/15/306 [7] DOI: 10.1016/j.mcm.2008.12.014 · Zbl 1171.15310 · doi:10.1016/j.mcm.2008.12.014 [8] Deng YP, J. Comput. Math. 23 pp 17– (2005) [9] DOI: 10.1016/j.laa.2008.03.019 · Zbl 1143.15011 · doi:10.1016/j.laa.2008.03.019 [10] DOI: 10.1109/72.914526 · doi:10.1109/72.914526 [11] Horn RA, Linear Multilinear Algebra [12] DOI: 10.1016/j.amc.2010.07.004 · Zbl 1204.15005 · doi:10.1016/j.amc.2010.07.004 [13] DOI: 10.1007/s10114-002-0204-8 · Zbl 1028.15011 · doi:10.1007/s10114-002-0204-8 [14] DOI: 10.1002/nla.701 · Zbl 1249.15020 · doi:10.1002/nla.701 [15] DOI: 10.1080/03081087408817070 · doi:10.1080/03081087408817070 [16] DOI: 10.1109/TSP.2006.870630 · Zbl 1373.94667 · doi:10.1109/TSP.2006.870630 [17] DOI: 10.1016/0024-3795(91)90063-3 · Zbl 0718.15006 · doi:10.1016/0024-3795(91)90063-3 [18] DOI: 10.1109/83.760310 · doi:10.1109/83.760310 [19] DOI: 10.1016/j.jfranklin.2007.05.002 · Zbl 1171.15015 · doi:10.1016/j.jfranklin.2007.05.002 [20] DOI: 10.1016/j.amc.2006.04.032 · Zbl 1109.65037 · doi:10.1016/j.amc.2006.04.032 [21] DOI: 10.1016/S0024-3795(02)00283-5 · Zbl 1023.93012 · doi:10.1016/S0024-3795(02)00283-5 [22] DOI: 10.1016/j.laa.2010.02.018 · Zbl 1205.15033 · doi:10.1016/j.laa.2010.02.018 [23] DOI: 10.1109/TSP.2008.2010600 · Zbl 1391.93261 · doi:10.1109/TSP.2008.2010600 [24] DOI: 10.1109/TSP.2010.2048323 · Zbl 1392.94488 · doi:10.1109/TSP.2010.2048323 [25] DOI: 10.1016/j.sigpro.2010.06.024 · Zbl 1203.94057 · doi:10.1016/j.sigpro.2010.06.024 [26] DOI: 10.1016/j.aml.2011.04.038 · Zbl 1388.15009 · doi:10.1016/j.aml.2011.04.038 [27] DOI: 10.1016/0167-6911(87)90003-X · Zbl 0623.93028 · doi:10.1016/0167-6911(87)90003-X [28] DOI: 10.1016/j.laa.2008.05.031 · Zbl 1158.15010 · doi:10.1016/j.laa.2008.05.031 [29] DOI: 10.1016/j.laa.2006.01.027 · Zbl 1109.65034 · doi:10.1016/j.laa.2006.01.027 [30] DOI: 10.1016/S0024-3795(97)10099-4 · Zbl 0933.15024 · doi:10.1016/S0024-3795(97)10099-4 [31] Yuan SF, Electron. J. Linear Algebra 23 pp 257– (2012) [32] DOI: 10.1016/0024-3795(95)00543-9 · Zbl 0873.15008 · doi:10.1016/0024-3795(95)00543-9 [33] DOI: 10.1016/j.laa.2006.08.004 · Zbl 1117.15017 · doi:10.1016/j.laa.2006.08.004 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.