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A real quaternion matrix equation with applications. (English) Zbl 1317.15016
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.

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.)
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References:
[1] DOI: 10.1016/j.sigpro.2004.04.001 · Zbl 1154.94331
[2] DOI: 10.1109/70.127239
[3] DOI: 10.1137/S0895479895270963 · Zbl 0912.93027
[4] Chu DL, SIAM J. Matrix Anal. Appl. 3 pp 1187– (2009)
[5] DOI: 10.1016/S0024-3795(99)00108-1 · Zbl 0959.93032
[6] DOI: 10.1088/0305-4470/33/15/306 · Zbl 0954.81008
[7] DOI: 10.1016/j.mcm.2008.12.014 · Zbl 1171.15310
[8] Deng YP, J. Comput. Math. 23 pp 17– (2005)
[9] DOI: 10.1016/j.laa.2008.03.019 · Zbl 1143.15011
[10] DOI: 10.1109/72.914526
[11] Horn RA, Linear Multilinear Algebra
[12] DOI: 10.1016/j.amc.2010.07.004 · Zbl 1204.15005
[13] DOI: 10.1007/s10114-002-0204-8 · Zbl 1028.15011
[14] DOI: 10.1002/nla.701 · Zbl 1249.15020
[15] DOI: 10.1080/03081087408817070
[16] DOI: 10.1109/TSP.2006.870630 · Zbl 1373.94667
[17] DOI: 10.1016/0024-3795(91)90063-3 · Zbl 0718.15006
[18] DOI: 10.1109/83.760310
[19] DOI: 10.1016/j.jfranklin.2007.05.002 · Zbl 1171.15015
[20] DOI: 10.1016/j.amc.2006.04.032 · Zbl 1109.65037
[21] DOI: 10.1016/S0024-3795(02)00283-5 · Zbl 1023.93012
[22] DOI: 10.1016/j.laa.2010.02.018 · Zbl 1205.15033
[23] DOI: 10.1109/TSP.2008.2010600 · Zbl 1391.93261
[24] DOI: 10.1109/TSP.2010.2048323 · Zbl 1392.94488
[25] DOI: 10.1016/j.sigpro.2010.06.024 · Zbl 1203.94057
[26] DOI: 10.1016/j.aml.2011.04.038 · Zbl 1388.15009
[27] DOI: 10.1016/0167-6911(87)90003-X · Zbl 0623.93028
[28] DOI: 10.1016/j.laa.2008.05.031 · Zbl 1158.15010
[29] DOI: 10.1016/j.laa.2006.01.027 · Zbl 1109.65034
[30] DOI: 10.1016/S0024-3795(97)10099-4 · Zbl 0933.15024
[31] Yuan SF, Electron. J. Linear Algebra 23 pp 257– (2012)
[32] DOI: 10.1016/0024-3795(95)00543-9 · Zbl 0873.15008
[33] DOI: 10.1016/j.laa.2006.08.004 · Zbl 1117.15017
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