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Similarity and other spectral relations for symmetric cones

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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

Contact: sturm@cauchy.crl.mcmaster.ca