Monotonic Properties of the Least Squares Mean

We settle an open problem of several years standing by showing that the least-squares mean for positive definite matrices is monotone for the usual (Loewner) order. Indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature. Our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity. Moreover, we introduce a weighted least squares means and extend our results to this setting.Comment: 21 page

On Ando–Li–Mathias geometric mean equations

AbstractIn this paper we consider a family of nonlinear matrix equations based on the higher-order geometric means of positive definite matrices that proposed by Ando–Li–Mathias. We prove that the geometric mean equationX=B+G(A1,A2,…,Am,X,X,…,X︸n)has a unique positive definite solution depending continuously on the parameters of positive definite Ai and positive semidefinite B. It is shown that the unique positive definite solutions Gn(A1,A2,…,Am) for B=0 satisfy the minimum properties of geometric means, yielding a sequence of higher-order geometric means of positive definite matrices

Self-scaled barriers for irreducible symmetric cones

Self-scaled barrier functions are fundamental objects in the theory of interior-point methods for linear optimization over symmetric cones, of which linear and semidefinite programming are special cases. We are classifying all self-scaled barriers over irreducible symmetric cones and show that these functions are merely homothetic transformations of the universal barrier function. Together with a decomposition theorem for self-scaled barriers this concludes the algebraic classification theory of these functions. After introducing the reader to the concepts relevant to the problem and tracing the history of the subject, we start by deriving our result from first principles in the important special case of semidefinite programming. We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras.Comment: 12 page

Jordan Algebras and Lie Semigroups.

For a Euclidean Jordan algebra V with the corresponding symmetric cone $\Omega$, we consider the semigroup \Gamma\sb{\Omega} of elements in the automorphism group G(T\sb{\Omega}) of the tube domain $V$ + $i\Omega$ which can be extended to $\Omega$ and maps $\Omega$ into itself. A study of this semigroup was first worked out by Koufany in connection to Jordan algebra theory and Lie theory of semigroups. In this work we give a new proof of Koufany\u27s results and generalize up to infinite dimensional Jordan algebras, so called $JB$-algebras. One of the nice examples of the semigroup \Gamma\sb{\Omega} is from the Jordan algebra Sym(n,\IR) of symmetric matrices. However, V\sb{\sigma} the set of all self-adjoint operators on \IR\sp{n} with respect to a non-degenerated symmetric bilinear form $\sigma$, is a non-Euclidean Jordan algebra with a cone \Omega\sb{\sigma} which is isomorphic to $\Omega$ of the symmetric cone of Sym(n,\IR). We get an isomorphism of the automorphism groups between two tube domains which also induces an isomorphism between two Lie semigroups. The Lorentzian cone, which is one of the irreducible symmetric cones, is an essential tool in the study of semigroups in Mobius and Lorentzian geometry. J. D. Lawson studied the Mobius and Lorentzian semigroups with an Ol\u27shanskii decomposition even in the infinite dimensional cases. We study these semigroups via a Jordan algebra theory