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Can anyone see a good way to visualize the SPD cone for 3x3 symmetric matrices? I'm interested in something that would highlight it's special structure, like non-smoothness.

Here's one attempt, looks pretty smooth to me

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I think the "symmetric" projections to three dimensions do a pretty good job. If you plot against the diagonal entries, $\begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}$, you get three nonsmooth edges along the coordinate axes. With the off-diagonal entries, $\begin{bmatrix}1 & x & y \\ x & 1 & z \\ y & z & 1\end{bmatrix}$, you get four nonsmooth vertices in a tetrahedral shape at coordinates $(\pm1,\pm1,\pm1)$.

enter image description here $\quad$ enter image description here

Any point at which the boundary is nonsmooth corresponds to a matrix with more than one vanishing eigenvalue. This is easy to check for the nonsmooth points in both these particular cases.

(P.S. For the benefit of future searchers, I think the phrase "cone of positive semidefinite matrices" should appear somewhere on this page :) ...)

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