Multilinear eigenvalue problem

In summary, the conversation discusses the problem of solving an eigenvalue problem with a rank-4 tensor A, an unknown eigenvalue lambda, and a set of unknown coefficients c_i. The speaker expresses their uncertainty about where to start and asks for any pointers or knowledge of the common name for this type of problem. They also mention their unsuccessful attempts to find relevant information through Google searches.
  • #1
Manchot
473
4
I'm trying to do something that requires solving an eigenvalue problem of the form
[tex]A_{imkl} c_m c_k c^*_l=\lambda c_i[/tex]
where A is a known rank-4 tensor, [itex]\lambda[/itex] is the eigenvalue, and the [itex]c_i[/itex]'s are a set of unknown coefficients that I need to determine. I would guess that this type of problem should be solvable, but I have no idea where to start. For all I know, it's a problem with a common name that I just don't know. Any pointers?
 
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  • #2
If anyone even knew what this class of problem is called, that would be tremendously helpful. (A Google search on various terms related to "eigenvalues" comes up with a lot of useless results.)
 

Related to Multilinear eigenvalue problem

What is a multilinear eigenvalue problem?

A multilinear eigenvalue problem is a mathematical problem that involves finding the eigenvalues and eigenvectors of a multilinear map. This means finding the values and corresponding vectors that satisfy the equation Ax = λx, where A is a multilinear map and λ is an eigenvalue.

What is the difference between a multilinear eigenvalue problem and a standard eigenvalue problem?

The main difference between a multilinear eigenvalue problem and a standard eigenvalue problem is that in a multilinear problem, the map A is multilinear, meaning it takes in multiple inputs and outputs a single value. In a standard eigenvalue problem, the map is linear, meaning it takes in a single input and outputs a single value.

What are some applications of multilinear eigenvalue problems?

Multilinear eigenvalue problems have many applications in fields such as physics, engineering, and computer science. They can be used to solve problems involving multidimensional data, such as image processing, signal processing, and data compression. They are also used in quantum mechanics, where they help determine the energy levels and properties of quantum systems.

How do you solve a multilinear eigenvalue problem?

Solving a multilinear eigenvalue problem involves finding the eigenvalues and eigenvectors of the multilinear map A. This can be done using various numerical methods, such as the power method, the QR algorithm, or the Jacobi method. These methods involve iterative processes that converge to the desired eigenvalues and eigenvectors.

What are some challenges in solving multilinear eigenvalue problems?

One of the main challenges in solving multilinear eigenvalue problems is that the dimensionality of the problem may be very high, making it computationally expensive. Another challenge is that there may be multiple eigenvalues and eigenvectors, making it difficult to determine the correct ones. Additionally, the accuracy of the solutions may be affected by numerical errors and round-off errors.

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