Singular value inequalities for functions of matrices

Abstract

In this paper, we prove several singular value inequalities for functions of matrices. As special cases of our results, we give some applications involving the spectral norms and numerical radii of matrices. Among other results, we prove that if A and B are n × n complex matrices and f is an increasing concave function such that f(0)=0, then for a,b≥0, we have

s_{j}(f(|aA^{∗}B+bB^{∗}A|))

≤ s_{i}(f(((a|A|²+b|B|²)/2))⊕f(((b|A|²+a|B|²)/2)))
+s_{j-i+1}(f(((bA^{∗}B+aB^{∗}A)/2))⊕f(((aA^{∗}B+bB^{∗}A)/2))),

for 1≤i≤j≤n. A special case of this inequality is related to recent inequalities given in <cite>BK</cite> and <cite>HK2007</cite>. Also, we prove that

‖Re A‖≤(1/2)(‖A‖+w(A))≤‖A‖.

Here, Re T, s_{j}(T), ‖T‖, and w(T) are the real part, the j^{th} singular value, the spectral norm, and the numerical radius of the matrix T, respectively.