Generalized Sunder Inequality

Abstract

V. Sunder proved that for n×n complex matrices A and B, with A being Hermitian and B being skew Hermitian with eigenvalues $\mathop{\left\{ \alpha_i \right\}}\nolimits^n_{i=1}$ and $\mathop{\left\{ \beta_i \right\}}\nolimits^n_{i=1}$ respectively (counting multiplicity) such that

$$\begin{array}{rrr}|\alpha_1|\geq\cdots\geq|\alpha_n|, \cr |\beta_1|\leq\cdots\leq|\beta_n|\end{array}$$

then

$$|\alpha_{i}-\beta{i}|\leq\|A-B\|$$

where $\|\cdot\|$ is the operator bound norm. We generalize Sunder’s result to the case of an m-tuple of n × n complex matrices, using the Clifford operator.

Mathematics Subject Classification (2010). Primary: 15A42; secondary 15A18.