Arnold's seminar
December 8th, 1998

N.V.Ilyushechkin
On some identities for the elements of a symmetric matrix.

A symmetric operator in R^n equipped with the standard Euclidean structure
is said to be regular, if it has no multiple eigenvalues.

For a regular symmetric operator A:
   1. the space of all symmetric operators is represented as a direct sum 
of two subspaces, one of which consists of all functions of A.

For an arbitrary symmetric operator A:
   2. A formula for the volume of the parallelepiped generated by the 
powers of A is derived.
   3. The discriminant of the characteristic polynomial of A is represented
as a sum of squares of certain polynomials in the matrix elements.