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Mathematics Colloquium

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Date:  Monday, March 30, 2009

Title:  DEGENERATE MATRICES

Abstract:  A symmetric matrix is called degenerate by physicists if it has a multiple eigenvalue.Wigner and von Neumann have shown long ago that the degenerate matrices form a variety of codimension two in the space of all symmetric matrices.This explains the phenomenon of "avoidance of crossing". I will show that if A,B,C are nXn symmetric matrices, and n is congruent 2 mod 4, there always exist three real numbers a,b,c, not all zero, such that aA+bB+cC is degenerate.This has interesting applications to symmetric hyperbolic systems of PDE-s.such as the equations of crystal optics. Degenerate matrices are characterized by the single equation discr[S]=0, where discr[S] is the discriminant of S.I shall present a new proof of the proposition that the discriminant can be represented as a sum of squares.


Speaker:  Peter Lax
Institution:  Courant Institute

 

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