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Exercise 2.64 #4
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How do you ensure |
Yes, it's not clear. Well, anyway, we can scale all |
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The actual solution is not too hard (though I also struggled a bit).
Let$1 \le k \le m$ and denote
$H = span{\phi_1,...,\phi_m }$ ,
$H_k = span{\phi_1,...,\phi_{k-1}, \phi_{k+1},...,\phi_m }$ .$H$ has dimension $m$ and $H_k$ has dimension $m-1$ . Now denote $E_k = Proj(H) - Proj(H_k)$ , where $Proj(H)$ denotes the (ortho)projection operator on the subspace $H$ . Clearly, $E_k \ge 0$ and $E_k \neq 0$ .$E_{m+1} = I - E_1-...-E_m$ .$\langle \phi_i|E_k|\phi_i \rangle = \langle \phi_i|Proj(H)|\phi_i \rangle - \langle \phi_i|Proj(H_k)|\phi_i \rangle = 1 - \langle \phi_i|Proj(H_k)|\phi_i \rangle = 0$ if $i \neq k$ and $>0$ if $i=k$ . This is what we needed.
Because of linear independence, subspace
Also, denote
Now calculate
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