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 PDEs.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
