**Real symmetric matrices with multiple eigenvalues
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** Beresford Parlett (Berkeley)
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*Résumé.*
We describe "avoidance of crossing" and its explanantion by von Neumann and Wigner. We show Lax's criterion for degeneracy and discover matrices whose determinants give the discriminant of a given symmetric matrix. This yields a simple proof of the bound given by Ilyushechkin on the number of terms in the expansion of the discriminant as a sum of squares. We present the 3 X 3 case in detail.