How do I interpret this picture of a spectrahedron?

Asked Nov 13 '24 at 20:54

Active Nov 13 '24 at 20:54

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On the Wikipedia page on spectrahedron, it shows the following picture of a spectrahedron (I avoid using the term image since it seems to have special meaning in this context)

I understand everything on the wikipedia page and have a rudimentary understanding of positive semidefinite matrices and semidefinite programming.

But nothing seems to connect to this particular picture of a spectrahedron. Is this a 3D projection (image) of some hypothetical p.s.d. matrix? What do the x, y, z axes represent? Why does it look like a bloated, rounded tetrahedron?

asked Nov 13 '24 at 20:54

Rufus

If you search on MSE with the keyword "spectrahedron" or "spectrahedra" (in "Search on Mathematics"), you will get more than 60 "hits", some of them connected to your question. See also this document connecting spectrahedra and super-ellipsoids. – Jean Marie Nov 13 '24 at 21:41
Cynthia Vinzant, What is a... spectrahedron?, Notices of the AMS, Volume 61, Number 5, May 2014. – Rodrigo de Azevedo Jun 16 '25 at 18:04
Bernd Sturmfels, Spectrahedra, GAeL Lecture I on Convex Algebraic Geometry, June 2010. – Rodrigo de Azevedo Jun 16 '25 at 18:12

How do I interpret this picture of a spectrahedron?

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