Similarity and other spectral relations for symmetric cones

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 deenite 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 + Y X)=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. There are two symmetric cones that are widely used in almost any area of applied mathematics , namely the nonnegative orthant and the cone of positive semi-deenite matrices. In optimization for instance, one considers linear programming models with nonnegativity constraints, and semi-deenite programming models with positive semi-deeniteness constraints. Linear optimization problems over Lorentz cones have recently also gained in interest. The nonnegative orthant, the positive semi-deenite matrices and the Lorentz cone are special cases of symmetric cones 5, 6, 8]. We say that a symmetric cone is a primitive, if it cannot be written as a Cartesian product (in a certain basis) of two other symmetric cones. Thus, the nonnegative reals < + form a primitive. The other primitives are the Lorentz cone, cone of positive semi-deenite matrices, with real, complex or quaternion entries, cone of 3 3 positive semi-deenite matrices with octonion entries.