Another Simple Proof of a Theorem of Chandler Davis

In 1957, Chandler Davis proved that unitarily invariant convex functions on the space of hermitian matrices are precisely those which are convex and symmetrically invariant on the set of diagonal matrices. We give a simple perturbation theoretic proof of this result. (Davis’ argument was also very short, though based on completely different ideas). Consider an orthogonally invariant function f defined on the set of n × n symmetric matrices. Such a function has to factor through the spectrum: (1) f(M) = g ◦ λ(M), where g is a symmetric function: (2) g(λ1, . . . , λn) = g(λσ(1), . . . , λσ(n)), for any permutation σ. In the sequel we shall further assume that f is a C convex function, and under this assumption we shall show that such functions are precisely those decomposing as per Eq. (1), with convex g. The argument and the statement are identical for unitarily invariant functions of Hermitian matrices; in that setting the theorem was proved in [Davis57], by a completely different argument (Davis made no regularity assumption, but this is easily dispensed with (see Remark 0.2)). To show this, let M = P + tQ, and let f̃P,Q(t) = f(M). It is enough to show that for any symmetric P,Q, df̃P,Q dt2 (0) > 0. We compute (dropping the subscript, since from now on P,Q do not vary):