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Let $O_n(\mathbb{R})$ act on $Sym_n(\mathbb{R})$, the symmetric matrices with real entries, via $S \mapsto A^T SA$ for $A \in O_2(\mathbb{R})$ and $Sym_n(\mathbb{R})$. What is the space of orbits $O_n(\mathbb{R})/Sym_n(\mathbb{R})$ as a set (and what is a basis for the topology)?

I know that we can diagonalize a symmetric matrix $A$ with $Q\in O_n(\mathbb{R})$ such that $QSQ^{-1}$ is diagnonal but I don't know how to continue. Thanks a lot for your help!

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  • $\begingroup$ Do you mean the set of orbits $Sym_2(\mathbb{R})/O_2(\mathbb{R})$? $\endgroup$ – Omnomnomnom Jan 17 '17 at 21:16
  • $\begingroup$ yes I do, added. $\endgroup$ – user657483 Jan 17 '17 at 21:19
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Each orbit contains a diagonal matrix of the form $$ \pmatrix{\lambda_1\\&\lambda_2} $$ with $\lambda_1 \geq \lambda_2$. This gives us a homeomorphism between $Sym_2(\mathbb{R})/O_2(\Bbb R)$ and $\Bbb R/ [(x,y) \sim (y,x)]$. That is, $Sym_2(\mathbb{R})/O_2(\Bbb R)$ is homeomorphic to the set of unordered real pairs.


We have the following basis for $\Bbb R/ [(x,y) \sim (y,x)]$: for every $(x,y)$ with $x \neq y$, take every neighborhood of $(x,y)$ in $\Bbb R$ which is small enough to miss the line $y = x$. For every point $(x,x)$, consider the usual neighborhoods, but exclude any points $(x,y)$ for which $x < y$.

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  • $\begingroup$ Thank you, does this statement generalize to unordered $n$-tuples of the reals? $\endgroup$ – user657483 Jan 17 '17 at 21:28
  • $\begingroup$ Yes, it generalizes exactly in that way. Interestingly, the resulting space is non-orientable. In the $2 \times 2$ case, I think you have something homeomorphic to an open Mobius strip. $\endgroup$ – Omnomnomnom Jan 17 '17 at 21:31
  • $\begingroup$ Can you give a basis of the topology of the orbit space? $\endgroup$ – user657483 Jan 17 '17 at 21:33
  • $\begingroup$ And: Can one say that the that symmetric matrices of the same eigenvalues are identified in this orbit space? $\endgroup$ – user657483 Jan 17 '17 at 21:35
  • $\begingroup$ To that last bit: yes, that's exactly what we're saying $\endgroup$ – Omnomnomnom Jan 17 '17 at 21:39
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Every orbit contains exactly one diagonal matrix $$\begin{pmatrix} a & 0 \\ 0 & b\end{pmatrix},$$ with $a\geq b,$ so the quotient space is exactly the set of pairs $(x, y)\in \mathbb{R}^2,$ with $x\geq y.$ Notice that the answer would be different for quotient by $SO(2),$ since then you can't sort, because the matrix $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ has determinant $-1.$

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  • $\begingroup$ Can one generalize this result for arbitrary $n$? $\endgroup$ – user657483 Jan 17 '17 at 21:23
  • $\begingroup$ Thanks for your answer and the very useful remark on sorting with $\begin{pmatrix}0&1\\1&0\end{pmatrix}\in O(n)$. $\endgroup$ – user657483 Jan 17 '17 at 22:01

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