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

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!

• 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). Commented Jul 6, 2018 at 9:38

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

• 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. Commented Jun 9, 2011 at 19:23

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.

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

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:

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