Hilbert-Schmidt Norm: Calculation & Solution

In summary, the conversation is about finding the norm of a defined function and discussing the integral set up and boundaries. The final result includes a characteristic function to ensure correctness.
  • #1
dirk_mec1
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  • #2
If the norm of blah is zero, then blah is zero. Is blah zero in this case?
 
  • #3
morphism said:
If the norm of blah is zero, then blah is zero. Is blah zero in this case?

You're right something is wrong.

But is the integral set up with the correct boundaries?
 
  • #4
Ok presuming the boundaries are ok I end up with:

[tex]||A||_{HS} = \frac{2 (b-a)^n}{((n-1)!)^2 (2n-1)(2n)} [/tex]

Is this correct?
 
  • #5
Did you remember to take the square root?
 
Last edited:
  • #6
You might want to include some characteristic function like [itex]\chi_{\{s\leq t\}}[/itex] in your kernel function.
 

1. What is the Hilbert-Schmidt norm?

The Hilbert-Schmidt norm is a mathematical concept used to measure the size or magnitude of a linear operator. It is often used in functional analysis and is defined as the square root of the sum of the absolute squares of the eigenvalues of the operator.

2. How do you calculate the Hilbert-Schmidt norm?

To calculate the Hilbert-Schmidt norm, you first need to find the eigenvalues of the linear operator. Then, you square each eigenvalue, add them together, and take the square root of the sum.

3. What is the significance of the Hilbert-Schmidt norm?

The Hilbert-Schmidt norm is important because it allows us to measure the size or magnitude of a linear operator and compare it to other operators. It also has applications in areas such as quantum mechanics and signal processing.

4. How is the Hilbert-Schmidt norm related to other norms?

The Hilbert-Schmidt norm is a specific type of norm, known as a Schatten norm. It is also closely related to the Frobenius norm, which is the square root of the sum of the squared elements of a matrix. In fact, the Hilbert-Schmidt norm is equivalent to the Frobenius norm for finite-dimensional matrices.

5. Can the Hilbert-Schmidt norm be used to solve problems?

Yes, the Hilbert-Schmidt norm can be used to solve various problems in mathematics and physics. For example, it can be used to find the best approximation of a given matrix or to determine the stability of a dynamical system. It is also used in optimization problems and in the study of differential equations.

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