Regarding the matrix inner product based on singular values, Lewis (1995) *"The convex analysis of unitarily invariant matrix functions"* states the result by von Neumann that $\langle X,Y \rangle \leq \langle \sigma\_X ,\sigma\_Y \rangle$. Does anyone know any easy proof or reference for it. I couldn't understand the reference cited in the paper.



Bound for matrix inner product based on singular values
For a "pedagogical" proof, see [A Note on von Neumann's Trace Inequality](http://page.math.tu-berlin.de/~grigo/10_Neumann_trace.pdf) by Rolf Dieter Grigorieff.



> 
> It has been remarked in the literature that *"unexpectedly, finding a
>  decent proof of this seemingly simple result turns out to be anything
>  but trivial"*. The aim of the present note is to present still a
>  further proof which seems to be elementary enough to correspond to the
>  simplicity of the statement in von Neumann's Theorem.
> 
> 
> 


The original proof by von Neumann is in "*Some matrix-inequalities and metrization of metric-space,*" Tomsk Univ. Rev. **1** (1937) 286-300 -- which I have not found online.


