Abstract :
Let Ai,Bi,Xi,i=1,…,m, be n-by-n matrices such that ∑mi=1|Ai|² and ∑mi=1|Bi|² are nonzero matrices and each Xi is positive semidefinite. It is shown that if f is a nonnegative increasing convex function on [0,∞) satisfying f(0)=0, then 2sj (f( |Σ^m_i=1 Ai*XiBi|/√||∑^m_i |Ai|² || ||∑^m_i|Bi|²||)) for j = 1,..., n. Applications of our results are given.
Keywords :
Singular value , arithmetic , geometric mean , direct sum , positive semidefinite matrix , convex function
