## The cone of positive semidefinite matrices is self-dual?  (reference needed)

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I'm seeking a reference for the following fact.

The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).

This result is relatively easy to prove, has been known for a long time, and is fundamental to things like semidefinite programming. Ideally, I would like a reference that reflects all three of those properties. Unfortunately, the properties themselves make it hard to find a good reference to cite. (Many sources I've looked at consider this result elementary and well-known enough to simply state without proof or reference. That was sort of my plan as well, but a referee is now asking for a reference, and seeing as how our paper is outside of optimization theory, I think that's probably reasonable.)

By the way, this result is occasionally referred to as Fejer's Trace Theorem, although I have never encountered an actual reference to any publication of Fejer. So if anyone knows the source of this attribution, that would be interesting.

Any help would be greatly appreciated!

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## 2 Answers

I am pretty sure Boyd's convex optimization (available on his web page as a pdf) talks about this (yes: example 2.24)

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Great!  I don't know if this is the ideal reference I'm after, but it certainly looks better than any I knew of before. – Louis Deaett Jun 9 2011 at 19:23

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Perhaps the Notes section of the classic book: Analysis on symmetric cones is of help.

In particular, they mention that the following paper of Koecher started the study of symmetric cones. I have not yet read this paper, so cannot say if it was this paper that described the self-duality result that you mention. But I hope the Notes section mentioned above does provide some clues.

M. Koecher (1957). Positivitätsbereiche in $R^n$. Amer J. Math., 79.

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I'm in the process of trying to get ahold of this book.  It sounds promising. – Louis Deaett Jun 9 2011 at 19:34
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