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On random and maximal cuts of the set of symmetric matrices with multiple eigenvalues

Research Group: 
Mathematical Analysis, Modelling, and Applications
Speaker: 
Khazhgali Kozhasov
Schedule: 
Friday, January 12, 2018 - 14:30 to 16:00
Location: 
A-133
Abstract: 
Let \Delta\subset \textrm{PSym}(n, \mathbb{R}) be the (projectivized) variety of real symmetric matrices with repeated eigenvalues. It is a well known fact that the codimension of \Delta is two. Let W\simeq \mathbb{R}\textrm{P}^2 be a generic projective 2-plane in \textrm{PSym}(n, \mathbb{R}). We prove the sharp upper bound:
 
\textrm{Card}(W\cap \Delta)\leq {n+1\choose 3}.

Moreover in the case when W is taken to be random and uniformly distributed in the Grassmannian \mathbb{G}(2, \textrm{PSym}(n, \mathbb{R}))  we prove that the expected number of intersection points equals:

 
\mathbb{E}\textrm{Card}(W\cap \Delta)={n\choose 2}.
The proof involves the computation of the volume of the variety \Delta, and this in turn is related to the probability that two of the eigenvalues of a GOE matrix are close.

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