Singular value inequalities with applications to norms and means of matrices

Abstract

In this paper, we obtain some upper bounds for the singular values of sums of product of matrices. The obtained forms involve direct sums and mean-like matrix quantities. As applications, several bounds will be found in terms of the Aluthge transform, matrix means, matrix monotone functions and accretive-dissipative matrices. For example, we show that if X is an \(n\times n\) accretive-dissipative matrix, then

$$\begin{aligned} {{s}_{j}}\left( X \right) \le \left( 1+\frac{\sqrt{2}}{2} \right) {{s}_{j}}\left( \Re X\oplus \Im X \right) , \end{aligned}$$

for \(j=1,2,\ldots n\), where \(s_j(\cdot ), \Re (\cdot )\) and \(\Im (\cdot )\) denote the \(j-\)th singular value, the real part and the imaginary part, respectively. We also show that if \(\sigma _f,\sigma _g\) are two matrix means corresponding to the operator monotone functions f, g, then

$$\begin{aligned} {{s}_{j}}\left( A{{\sigma }_{f}}B-A{{\sigma }_{g}}B \right) \le \left\| A \right\| {{s}_{j}}\left( f\left( {{A}^{-\frac{1}{2}}}B{{A}^{-\frac{1}{2}}} \right) \oplus g\left( {{A}^{-\frac{1}{2}}}B{{A}^{-\frac{1}{2}}} \right) \right) , \end{aligned}$$

for \(j =1,2, \ldots , n\), where A, B are two positive definite \(n\times n\) matrices.