From eigenvectors to eigentensors. Is eigentensor (hypermatrix) theory developed and useful?

Question

Just as eigenvectors are invariant vectors up to a multiplicative constant $Au=\lambda u$, I wondered if eigentensor theory is equally developed for any arbitray tensor or hypermatrix array, i.e., $AB=\lambda B$, where A and B are suitable hyperobjects, and, maybe, even $\lambda$ could be an eigentensor as well.

Related: Why should we encounter eigentensors naturally? Is the eigentensor/hypermatrix theory hard? Any nice reference for current status of the subject including something else beyond hyperdeterminants (Kapranov et alii)?

Answer 1

The answer to both questions is yes. The theory is very new, but here are two textbooks you could explore:

• Liqun Qi & Ziyan Luo - Tensor Analysis: Spectral Theory and Special Tensors (2017) -- A modern introduction to hypermatrices. It includes an entire chapter on eigenvalues.
• Liqun Qi & Haibin Chen & Yannan Chen - Tensor Eigenvalues and Their Applications (2018) -- This book will answer the question of why they are useful. It includes applications to diffusion tensor imaging, mechanics, and quantum physics.