Similarity and other spectral relations for symmetric cones

Abstract

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 2X∼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.