# When will the equality of Neumann's Trace inequality holds?

I am now studying the Neumann's trace inequality.

Some of the literature said that the equality holds when the two matrices have simultaneous ordered spectral decomposition.

Yet they don't really explain what is "simultaneous ordered spectral decomposition".

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Have a look at Theorem 0.1 of drhea.net/wp-content/uploads/2011/01/vonNeumann.pdf –  matt Aug 14 '12 at 11:01