SINGULAR VALUES OF CONVEX FUNCTIONS OF MATRICES

Let A(i), B-i, X-i, i = 1,..., m, be n-by-n matrices such that Sigma(m)(i=1) vertical bar A(i)vertical bar(2) and Sigma(m)(i=1) vertical bar B-i vertical bar(2) are nonzero matrices and each X-i is positive semidefinite. It is shown that if f is a nonnegative increasing convex function on [0, infinity) satisfying f (0) = 0, then 2s(j) (f(vertical bar Sigma(m)(i-1)A(i)*XiBi vertical bar/root parallel to Sigma(m)(i=1)vertical bar A(i)vertical bar(2)parallel to parallel to Sigma(m)(i=1)vertical bar B-i vertical bar(2)parallel to)) <= s(j) (circle plus(m)(i=1)f(2X(i))) for j = 1,..., n. Applications of our results are given.