Spectrahedra and Directional Derivatives of Determinants

on YouTube ↗32m

" summary

Starts from a definite determinantal polynomial and takes the directional derivative in the identity direction, producing the degree n−1 polynomial equal to the sum of (n−1)-by-(n−1) principal minors (equivalently the elementary symmetric polynomial of eigenvalues). The main claim is that this derivative cone is spectrahedral: construct a PSD linear matrix pencil by restricting to matrices with trace zero (analogue of V^T diag(x) V), proven geometrically and algebraically. Smallest unresolved case: 5×5 matrices, degree 3.

AI-written summary · about

" chapters (14)

1. 0:00Intro
2. 1:55Hyperbolic polynomials A polynomial p homogeneous of degreed in variables is hyperbolic with respect to ER if
3. 5:21Key examples
4. 7:59Hyperbolic programming
5. 9:58Derivative relaxations/Renegar derivatives
6. 12:43Examples: elementary symmetric polynomials
7. 16:07Generalized Lax conjecture
8. 18:35Lax-type problems for derivatives
9. 22:33(Projected) spectrahedral descriptions
10. 23:33Example: Sanyal's representation
11. 25:14Main result
12. 26:19Sketch of proof: "geometric"
13. 28:10Sketch of proof: algebraic Polynomial identity
14. 29:58Summary