Group invariance and convex matrix analysis

Abstract. Certain interesting classes of functions on a real inner product space are invari-ant under an associated group of orthogonM linear transformations. This invariance can be made explicit via a simple decomposition. For example, rotationally invariant functions on R2 are just even functions of the Euclidean norm, and functions on the Hermitian matrices (with trace inner product) which are invariant under unitary similarity transformations are just symmetric functions of the eigenvalues. We develop a framework for answering geometric and analytic (both classical and nonsmooth) questions about such a function by answering the corresponding question for the (much simpler) function appearing in the decomposition. The aim is to understand and extend the foundations of eigenvMue optimization, matrix approximation, and semidefinite programming. Key words, convexity, group invariance, nonsmooth analysis, semidefinite program, eigenvalu