Disproof of a conjecture

Summary

Let the symmetrical matrixA, of ordern, be defined in terms of then numbers x_j, by the formula It is known that, if the n numbers x_j are the n zeros of the Hermite polynomial of ordern,H _n(x)_j = 0, the matrix A has the eigenvalues 0, 1, 2, …,n - 1. It has been conjectured that the requirement that the matrix A have the first n nonnegative integers as its eigenvalues implies that the n numbers x_j coincide, up to a translation, with then zeroes of the Hermite polynomial of ordern. This conjecture is falsified, forn = 4.