Geodesic
On some identities for the elements of a~symmetric matrix
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 119-144

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Let $\operatorname{Sym}(n)$ be the space of $n$-dimensional real symmetric matrices, and let $X\in\operatorname{Sym}(n)$. The matrices $E,X,X^2,\dots,X^{n-1}$ can be regarded as vectors of Euclidean space of dimension $n^2$. Denote by $V(E,X,\dots,X^{n-1})$ the volume of the parallelepiped built on these vectors. It is proved that $$ V^2(E,X,\dots,X^{n-1})=D(X), $$ where $D(X)$ is the discriminant of the characteristic polynomial of the matrix $X$. Two classes of smooth maps of the space $\operatorname{Sym}(n)$ are described. 
@article{ZNSL_2003_303_a6,
     author = {N. V. Ilyushechkin},
     title = {On some identities for the elements of a~symmetric matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {119--144},
     publisher = {mathdoc},
     volume = {303},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_303_a6/}
}
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N. V. Ilyushechkin. On some identities for the elements of a~symmetric matrix. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 119-144. http://geodesic.mathdoc.fr/item/ZNSL_2003_303_a6/