Abstract
The principal aim of this note is to establish an effectively self-contained proof of J. von Neumann's inequality $$\left| {tr\left( {AB} \right)} \right| \leqslant \sum\limits_{r = 1}^n {\varrho _r \sigma _r } $$ , whereA, B are any complexn×n matrices with singular values ϱ1⩾...⩾ϱ n , σ1⩾...⩾σ n respectively.
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