Von Neumann's trace inequality

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, von Neumann's trace inequality, named after its originator John von Neumann, states that for any n × n complex matrices AB with singular values \alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n and \beta_1 \ge \beta_2 \ge \cdots \ge \beta_n respectively,[1]

\left| \mathrm{trace}(AB) \right| \le \sum_{i=1}^n \alpha_i \beta_i.

The equality is achieved when A and B are simultaneously unitarily diagonalizable. (See Trace (linear algebra).)


See also[edit]