Professor D. Lax
COURANT INSTITUTE OF MATHEMATICAL SCIENCES, NYU
Degenerate Symmetric Matrices
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".On the other hand the degenerate matrices are characterised by the single equation discr(S)=0, where discr(S) is the discriminant of S.In this talk we investigate the nature of the discriminant, especially its representation as a sum of squares. In the second part it will be shown that some pencils of real symmetric matrices always contain a degenerate one.