Von Neumann's trace inequality

In mathematics, von Neumann's trace inequality, named after its originator John von Neumann, states that for any n × n complex matrices A, B 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).)

References

↑ Mirsky 1975.

Mirsky, L. (1975), "A trace inequality of John von Neumann", Monatsh. Math. 79 (4): 303–306, doi:10.1007/BF01647331, MR 0371930

Von Neumann's trace inequality

References

See also