Similarity and other spectral relations for symmetric cones

Jos F. Sturm

A one--to--one relation is established between the nonnegative spectral values of a vector in a primitive symmetric cone and the eigenvalues of its quadratic representation. This result is then exploited to derive similarity relations for vectors with respect to a general symmetric cone. For two positive definite matrices $X$ and $Y$, the square of the spectral geometric mean is similar to the matrix product $XY$, and it is shown that this property carries over to symmetric cones. 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 that are useful in the context of interior point methods, and an extension of Stein's theorem to symmetric cones. Key words. Symmetric cone, Euclidean Jordan algebra, optimization.

Communications Research Lab., McMaster University, 1280 Main Street West, HAMILTON , ONTARIO L8S 4K1, CANADA July 1998