Eigenvalues of a compact positive definite operator

In summary, to show that the sum of the inner products between A and an orthonormal set ψ1,...,ψn is less than or equal to the sum of the eigenvalues of A in decreasing order, you can use the fact that the first eigenvalue is the supremum of the inner product of A with a unit vector. By using induction, you can prove this for all n.
  • #1
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eigenvalues of a compact positive definite operator!

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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  • #2


Both the left and right expression look like tr(A).
 
  • #3


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?
 

1. What are eigenvalues of a compact positive definite operator?

The eigenvalues of a compact positive definite operator are the set of scalars that, when multiplied by the corresponding eigenvectors, result in the original vector being scaled by a certain amount. These eigenvalues are always positive and are important in understanding the behavior of the operator.

2. How do eigenvalues relate to the spectrum of a compact positive definite operator?

The spectrum of a compact positive definite operator is the set of all possible eigenvalues. In other words, the spectrum is the range of values that the eigenvalues can take on. This is important in understanding the overall properties of the operator.

3. Can the eigenvalues of a compact positive definite operator be negative?

No, the eigenvalues of a compact positive definite operator are always positive. This is because the operator is defined to be positive definite, meaning that it has certain properties that ensure all of its eigenvalues are positive.

4. How can eigenvalues be calculated for a compact positive definite operator?

The eigenvalues of a compact positive definite operator can be calculated using various numerical methods, such as the power method or the QR algorithm. These methods involve iterative processes that converge to the eigenvalues of the operator.

5. What is the significance of the eigenvalues of a compact positive definite operator in applications?

The eigenvalues of a compact positive definite operator are important in many applications, including data analysis, signal processing, and differential equations. They can provide insight into the behavior and characteristics of the operator, which can be used to make informed decisions and solve problems efficiently.

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