What Is the Multilinear Eigenvalue Problem Called?

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SUMMARY

The problem discussed is known as the Multilinear Eigenvalue Problem, specifically involving a rank-4 tensor denoted as A_{imkl}. The equation A_{imkl} c_m c_k c^*_l = λ c_i represents the relationship between the tensor, the unknown coefficients c_i, and the eigenvalue λ. This type of eigenvalue problem is less common and often requires specialized numerical methods for solutions. Identifying this problem correctly is crucial for applying the appropriate mathematical techniques and tools.

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  • Understanding of multilinear algebra and tensor operations
  • Familiarity with eigenvalue problems and their mathematical formulations
  • Knowledge of numerical methods for solving eigenvalue problems
  • Experience with programming languages or software that handle tensor computations, such as MATLAB or Python with NumPy
NEXT STEPS
  • Research numerical methods for solving Multilinear Eigenvalue Problems
  • Explore tensor decomposition techniques relevant to rank-4 tensors
  • Learn about specialized libraries for tensor computations in Python, such as TensorLy
  • Study applications of multilinear eigenvalue problems in fields like machine learning and physics
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Mathematicians, physicists, and engineers working with tensor analysis, as well as researchers involved in advanced numerical methods for eigenvalue problems.

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I'm trying to do something that requires solving an eigenvalue problem of the form
A_{imkl} c_m c_k c^*_l=\lambda c_i
where A is a known rank-4 tensor, \lambda is the eigenvalue, and the c_i'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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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.)
 

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