On the codimension of the variety of symmetric matrices with multiple eigenvalues

According to a result of Wigner and von Neumann, the dimension of the set $\mathcal M$ of $n\times n$ real symmetric matrices with multiple eigenvalues is equal to $N-2$, where $N=n(n+1)/2$. This value is determined by counting the number of free parameters in the spectral decomposition of a matrix. We show that the same dimension is obtained if $\mathcal M$ is interpreted as an algebraic variety.