Let Sym (n) be the space of n-dimensional real symmetric matrices and let X ∈ Sym (n). The matrices E, X, X~2,..., X~(n-1) may be considered as vectors of the Euclidean space of dimension n~2. Denote by V(E, X,... ,X~(n-1)) the volume of the parallelepiped spanned by these vectors. It is proved that V~2(E,X,...,X~(n-1))=D(X), where D(X) is the discriminant of the characteristic polynomial of the matrix X. Two classes of smooth maps of the space Sym (n) are described.
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