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Eigenvalues of a compact positive definite operator

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SVD
#1
Dec20-12, 09:03 AM
P: 6
Let A be a compact positive definite operator on Hilbert space H.
Let ψ1,...ψn be an orthonormal set in H.
How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where
λ1≥λ2≥λ3≥..... be the eigenvalues of A in decreasing order.
Can someone give me a hint???
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mfb
#2
Dec20-12, 10:40 AM
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P: 11,928
Both the left and right expression look like tr(A).
micromass
#3
Dec20-12, 12:16 PM
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P: 18,334
Try induction.

Do you know that

[tex]\lambda_1=sup\{<Ax,x>~\vert~x\in H,~\|x\|=1\}[/tex]

??

If you know this, then the case n=1 should be easy. Can you find an argument to deal with the other cases?


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