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Similarity and other spectral relations for symmetric cones. (English) Zbl 0973.90093
Summary: The similarity relations that are derived in this paper reduce to well-known results in the special case of symmetric matrices. In particular, for two positive definite matrices $$X$$ and $$Y$$, the square of the spectral geometric mean is known to be similar to the matrix product $$XY$$. It is shown in this paper that this property carries over to symmetric cones. More elementary similarity relations, such as $$XY^2 X\sim YX^2Y$$, are generalized as well. We also extend the result that the eigenvalues of a matrix product $$XY$$ are less dispersed than the eigenvalues of the Jordan product $$(XY+YX)/2$$. The paper further contains a number of inequalities on norms and spectral values; this type of inequality is often used in the analysis of interior point methods (in optimization). We also derive an extension of Stein’s theorem to symmetric cones.

##### MSC:
 90C46 Optimality conditions and duality in mathematical programming 15B48 Positive matrices and their generalizations; cones of matrices 17C99 Jordan algebras (algebras, triples and pairs)
##### Software:
CSDP; SeDuMi; SDPpack; SDPT3
Full Text:
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