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Common Hermitian least squares solutions of matrix equations \(A_1XA^*_1=B_1\) and \(A_2XA^*_2=B_2\) subject to inequality restrictions. (English) Zbl 1231.15002
Summary: We give some closed-form formulas for calculating maximal and minimal ranks and inertias of \(P - X\) with respect to \(X\), where \(P\in C_H^n\) is given, \(X\) is a common Hermitian least squares solutions to the matrix equations \(A_1XA^*_1=B_1\) and \(A_2XA^*_2=B_2\). As an application, we derive necessary and sufficient conditions for \(X>P(\geq P,<P,\leq P)\) in the Löwner partial ordering. In addition, we give identifying conditions for the existence of definite common Hermitian least squares solutions to matrix equations \(A_1XA^*_1=B_1\) and \(A_2XA^*_2=B_2\).

MSC:
15A03 Vector spaces, linear dependence, rank, lineability
15A24 Matrix equations and identities
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[1] Marsaglia, G.; Styan, G.P.H., Equalities and inequalities for ranks of matrices, Linear multilinear algebra, 2, 269-292, (1974)
[2] Tian, Y., Equalities and inequalities for inertias of Hermitian matrices with applications, Linear algebra appl., 433, 263-296, (2010) · Zbl 1205.15033
[3] Baksalary, J.K., Nonnegative definite and positive definite solutions to the matrix equation \(A X A^\ast = B\), Linear multilinear algebra, 16, 133-139, (1984) · Zbl 0552.15009
[4] Dai, H.; Lancaster, P., Linear matrix equations from an inverse problem of vibration theory, Linear algebra appl., 246, 31-47, (1996) · Zbl 0861.15014
[5] Groß, J., A note on the general Hermitian solution to \(A X A^\ast = B\), Bull. Malaysian math. soc. (second series), 21, 57-62, (1998) · Zbl 1006.15011
[6] Groß, J., Nonnegtive-definite and positive-definite solution to the matrix equation \(A X A^\ast = B\) revisited, Linear algebra appl., 321, 123-129, (2000)
[7] Khatri, C.G.; Mitra, S.K., Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. appl. math., 31, 579-585, (1976) · Zbl 0359.65033
[8] Liu, Y.; Tian, Y.; Takane, Y., Ranks of Hermitian and skew-Hermitian solutions to the matrix equation \(A X A^\ast = B\), Linear algebra appl., 431, 2359-2372, (2009) · Zbl 1180.15018
[9] Wei, M.; Wang, Q., On rank-constrained Hermitian nonnegtive-definite least squares solutions to the matrix equation \(A X A^\ast = B\), Int. J. comput. math., 84, 945-952, (2007) · Zbl 1129.15012
[10] Zhang, X.; Cheng, M., The rank-constrained Hermitian nonnegtive-definite and positive-definite solutions to the matrix equation \(A X A^\ast = B\), Linear algebra appl., 370, 163-174, (2003) · Zbl 1026.15011
[11] Tian, Y., Maximization and minimization of the rank and inertia of the Hermitian matrix expression \(A - B X -(B X)^\ast\) with applications, Linear algebra appl., 434, 2109-2139, (2011) · Zbl 1211.15022
[12] Zhang, X., The general common Hermitian nonnegative definite solution to the matrix equations \(A X A^\ast = B B^\ast\) and \(C X \mathbb{C}^\ast = D D^\ast\) with applications in statistic, J. multivariate anal., 93, 257-266, (2005)
[13] Tian, Y.; Liu, Y., Extremal ranks of some symmetric matrix expressions with applications, SIAM J. matrix anal. appl., 28, 890-905, (2006) · Zbl 1123.15001
[14] Liu, Y.; Tian, Y., MAX-MIN problems on the ranks and inertias of the matrix expressions \(A - B X C \pm(B X C)^\ast\) with applications, J. optim. theory appl., 148, 593-622, (2011) · Zbl 1223.90077
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