Kippenhahn's Theorem in Higher Dimensions
summary
Kippenhahn's theorem identifies a matrix numerical range as the convex hull of the projective dual curve coming from the determinantal hyperbolic polynomial; for a 4×4 example the boundary quartic produces a degree 12 dual. In higher dimensions the convex hull of the joint numerical range equals the convex dual of the associated spectrahedron only after one excludes lower‑dimensional singular pieces of the dual variety (lines, isolated real nodes). Tools: hyperbolic polynomials, determinantal representations, projective duality and convex duality; the method succeeds once singular loci are removed.
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chapters (9)
- 0:00Intro
- 0:42The numerical range
- 7:17Kippenhahn's Theorem
- 13:10Duality for plane curves
- 17:02Duality for hyperbolic curves
- 25:48The joint numerical range
- 32:19Projective duality in higher dimensions
- 36:41Famous example
- 41:41How to fix it
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