# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.