Geodesic
On some classes of smooth transformations in the space of symmetric matrices
Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 75-86 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\mathrm{Sym}(n)$ be the space of $n$-dimensional real symmetric matrices. Two families of infinitely smooth transformations in $\mathrm{Sym}(n)$ are considered. First, the family of transformations $$ {\mathcal F}\colon\operatorname{Sym}(n)\to\operatorname{Sym}(n), $$ having the following property: for any matrix $X\in\mathrm{Sym}(n)$ and an orthogonal matrix $C$ such that $C^{-1}XC$ is a diagonal matrix, $C^{-1}\mathcal{F}(X)C$ is also a diagonal matrix. Second, the family of transformations $$ {\mathcal G}\colon\operatorname{Sym}(n)\to\operatorname{Sym}(n), $$ such that the diagonal entries of the matrix $C^{-1}\mathcal{G}(X)C$ are zero whenever the matrix $C^{-1}XC$ is diagonal. 
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N. V. Ilyushechkin. On some classes of smooth transformations in the space of symmetric matrices. Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 75-86. http://geodesic.mathdoc.fr/item/SM_1995_80_1_a3/

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