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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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  • $\begingroup$ The claim is proved on p. 9 of the book of Faraut and Koranyi. It is sufficiently elementary and well-known as to need no reference.The book of Nesterov and Nemirovskii Interior-Point Polynomial Algorithms in Convex Programming has some discussion of this cone in the context of optimization (see e.g. section 5.4.5, but it appears elsewhere in the book too). $\endgroup$
    – Dan Fox
    Commented Jul 6, 2018 at 9:38

3 Answers 3

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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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  • $\begingroup$ 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. $\endgroup$ Commented 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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  • $\begingroup$ I'm in the process of trying to get ahold of this book. It sounds promising. $\endgroup$ Commented Jun 9, 2011 at 19:34
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I know you are looking for a reference and you probably know how to prove it (and that the post is old). However, I want to include a short form of the proof for those coming to the post for this reason.

Note that

$$\def\<{\langle}\def\>{\rangle}\DeclareMathOperator{\tr}{tr} \<X,Y\>=\tr(XY)=\sum_{i,j=1}^n\lambda_i^X \lambda_j^Y\<v_i^X,v_j^Y\>^2,$$

where $\lambda_i^X$ is the the $i$-th eigenvalue of $X$ corresponding to the (normalized) eigenvector $v_i^X$ (and equivalently for $Y$). Consider the eigenvectors to be choosen as an orthonormal system. From this, it is easy to see $\<X,Y\>\ge 0$ for positive semi-definite matrices $X$ and $Y$. However, if e.g. $X$ is not positive semi-definite but has a negative eigenvalue, say $\lambda_1^X<0$, then we can choose

$$Y := v_1^X (v_1^X)^\top \in \mathbf S^n_+.$$

This gives $\<X,Y\><0$. And this is exactly what proves that $\mathbf S^n_+$ is self-dual:

$$X\in\mathbf S^n_+ \qquad\Longleftrightarrow \qquad \<X,Y\>\ge 0\text{ for all $Y\in\mathbf S^n_+$}.$$

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