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Two orderings on a convex cone of nonnegative definite matrices. (English) Zbl 0626.15008

Let \({\mathcal M}\) be a convex cone of real nonnegative definite matrices. For \(M_ 1,M_ 2\in {\mathcal M}\) denote \(M_ 1\geq M_ 2\) or \(M_ 1\succcurlyeq M_ 2\) if, respectively, \(M_ 1-M_ 2\) or \(M^ 2_ 1- M^ 2_ 2\) is nonnegative definite. Conditions for equivalence of the two orderings are given. In particular, if \(M_ 1M_ 2=M_ 2M_ 1\) for all \(M_ 1,M_ 2\in {\mathcal M}\), then the two orderings are equivalent on \({\mathcal M}\). The reverse direction of this result is posed as a conjecture.
Reviewer: S.Zlobec

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A09 Theory of matrix inversion and generalized inverses
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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References:

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