Stȩpniak, Czesław Two orderings on a convex cone of nonnegative definite matrices. (English) Zbl 0626.15008 Linear Algebra Appl. 94, 263-272 (1987). 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 Cited in 3 ReviewsCited in 4 Documents 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 Keywords:generalized inverse; convex cone; nonnegative definite matrices; orderings PDFBibTeX XMLCite \textit{C. Stȩpniak}, Linear Algebra Appl. 94, 263--272 (1987; Zbl 0626.15008) Full Text: DOI References: [1] Greville, T. N.E., Note on the generalized inverse of a matrix product, SIAM Rev., 8, 518-521 (1966) · Zbl 0143.26303 [2] Marshall, A. W.; Olkin, I., Inequalities: Theory of Majorization and its Applications (1979), Academic: Academic New York · Zbl 0437.26007 [3] Rao, C. R., Linear Statistical Inference and its Applications (1973), Wiley: Wiley New York · Zbl 0169.21302 [4] Rao, C. R.; Mitra, S. K., Generalized Inverse of Matrices and its Applications (1971), Wiley: Wiley New York [5] St \(ę\) pniak, C., Ordering of nonnegative definite matrices with application to comparison of linear models, Linear Algebra Appl., 70, 67-71 (1985) [6] St \(ę\) pniak, C.; Wang, S. G.; Wu, C. F.J., Comparison of linear experiments with known covariances, Ann. Statist., 12, 358-365 (1984) · Zbl 0546.62004 [7] Wu, C. F., On some ordering properties of the generalized inverses of nonnegative definite matrices, Linear Algebra Appl., 32, 49-60 (1980) · Zbl 0436.15004 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.