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

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

2011-05-27T18:09:42Z

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!

Answer by Igor Rivin for The cone of positive semidefinite matrices is self-dual? (reference needed)

2011-05-27T20:06:38Z

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

Answer by S. Sra for The cone of positive semidefinite matrices is self-dual? (reference needed)

2011-05-27T20:07:38Z

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.