A conjugacy class under O(n), orthogonal projection

  • Thread starter Sajet
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  • #1
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This is not really a homework question per se but I wasn't sure where else to put it:

In a script I'm reading the following set is defined:

[itex]P(n)_k := \{p \in S(n) | p^2 = p, \text{trace } p = k\}[/itex]

(i.e. the set of all real orthogonal projection matrices with trace k).

Now the following statement is made:

"[itex]P(n)_k[/itex] is a submanifold of the affine space [itex]S(n)_k[/itex] since it is the conjugacy class of the matrix [itex]p_0 = \begin{pmatrix}I_k && 0 \\ 0 && 0\\ \end{pmatrix}[/itex], i.e. the orbit of [itex]p_0[/itex] under the action of the group O(n) on S(n) by conjugation."

([itex]S(n)_k[/itex] is the set of all real symmetric matrices with trace k.)

I don't understand the second part of this statement. The conjugacy class should be:

[itex]\{Ap_0A^{-1} | A \in O(n)\} = \{Ap_0A^{t} | A \in O(n)\} =[/itex]

[itex] \{\begin{pmatrix}a_{11}^2 && 0 && ... && 0 && ... && 0 \\ 0 && a_{22}^2 && ... && 0 && ... \\ 0 && 0 && ... && a_{kk}^2 && ... && 0 \\ 0 && 0 && 0 && 0 && ... && 0\end{pmatrix} | (a_{ij}) \in O(n)\}[/itex]

and I don't see why this equals [itex]P(n)_k[/itex]. (Or maybe my calculation is wrong.)
 
Last edited:

Answers and Replies

  • #2
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I'm lost at how you found

[itex]Ap_0A^{t}=\begin{pmatrix}a_{11}^2 && 0 && ... && 0 && ... && 0 \\ 0 && a_{22}^2 && ... && 0 && ... \\ 0 && 0 && ... && a_{kk}^2 && ... && 0 \\ 0 && 0 && 0 && 0 && ... && 0\end{pmatrix}[/itex]

That doesn't seem correct at all.
 
  • #3
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You're right. I don't know what happened there. I'll take a look at this again.
 
  • #4
22,129
3,297
Some keywords which might help: "diagonalization of symmetric matrices"
 
  • #5
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Hey! First of all, thank you for your help! I didn't get back to this in the last couple of days but I will take a closer look tomorrow.
 
  • #6
48
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Ok, I figured this out. Thank you.

Just one more thing: I can't really follow why this makes the conjugacy class a submanifold. I know that certain kinds of orbit spaces are manifolds but this is not true for every group action, and I couldn't find anything on conjugacy classes always being submanifolds.
 

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