- #1
Sajet
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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.)
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.)
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