Geodesic
On Hermitian Nonnegative-Definite Solutions to Matrix Equations
Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 470-475

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For a system of $q$ matrix equations denoted by $$ \mathbf A_i\mathbf X\mathbf A_i^*=\mathbf B_i\mathbf B_i^*,\qquad i=1,2,\dots,q, $$ the problem of the existence of Hermitian nonnegative-definite solutions is considered in this note. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with $q=2$ to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who provided a revision of Young et al. [2]. Moreover, we give a necessary condition for the general case and then pose a conjecture, for which at least some special situations are argued.
Mots-clés : matrix equation
Keywords: Hermitian nonnegative-definite solution, Hermitian matrix, Moore–Penrose inverse. 
@article{MZM_2009_85_3_a14,
     author = {X.-Q. Liu and J.-Y. Rong},
     title = {On {Hermitian} {Nonnegative-Definite} {Solutions} to {Matrix} {Equations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {470--475},
     publisher = {mathdoc},
     volume = {85},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a14/}
}
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X.-Q. Liu; J.-Y. Rong. On Hermitian Nonnegative-Definite Solutions to Matrix Equations. Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 470-475. http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a14/