Open AccessGENERALIZATION OF A THEOREM BY JOHN VON NEUMANN ON THE TRACE OF CERTAIN MATRIX PRODUCTS 1
Author(s) -
Kristof Walter
Publication year - 1969
Publication title -
ets research bulletin series
Language(s) - English
Resource type - Journals
eISSN - 2333-8504
pISSN - 0424-6144
DOI - 10.1002/j.2333-8504.1969.tb00744.x
Subject(s) - trace (psycholinguistics) , von neumann architecture , mathematics , generalization , matrix (chemical analysis) , diagonal , diagonal matrix , unitary state , product (mathematics) , von neumann algebra , pure mathematics , unitary matrix , combinatorics , mathematical analysis , geometry , philosophy , linguistics , materials science , political science , law , composite material
In 1937, John von Neumann gave a theorem on the maximum of the real part of the trace of a matrix product Z 1 A 1 Z 2 A 2 where A 1 ,A 2 are fixed complex matrices and Z 1 ,Z 2 run independently over all unitary matrices. The theorem is extended to admit analogously tr Z 1 A 1 … Z n A n , n > 2. The Proof is not related to von Neumann's method. Four seemingly different, but in fact equivalent, versions of the result are given. These differences stem from initially requiring A j to be real diagonal as well as from considering separately the maxima of the absolute value, the real part and the imaginary part of tr Z 1 A 1 … Z n A n . An application is given by way of illustration.