Geodesic
On the codimension of the variety of symmetric matrices with multiple eigenvalues
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 34-46

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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. 
@article{ZNSL_2005_323_a3,
     author = {M. Dana and Kh. D. Ikramov},
     title = {On the codimension of the variety of symmetric  matrices with multiple eigenvalues},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {34--46},
     publisher = {mathdoc},
     volume = {323},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a3/}
}
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M. Dana; Kh. D. Ikramov. On the codimension of the variety of symmetric  matrices with multiple eigenvalues. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 34-46. http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a3/