Is there such a thing as an eigentensor?

[Mark R]

I'm familiar with the concept of taking the eigenvalues/vectors of a 2nd-order tensor.

But let's take a 4th order tensor. Is there a definition of eigenvalues/vectors for such a tensor? If there is, I can't work out what it might be.

Now, what if we rewrite the 4th order tensor in Voigt notation. Now, we can find the eigenvectors (which are now 6th dimensional). But what do they mean? Do these eigenvectors represent tensors in Voigt notation? In other words, could I rewrite the 6 dimensional vectors as symmetric 2nd-order tensors (eigentensors?)

 

[eLiu]

http://www.stat.uchicago.edu/~lekheng/work/camsap.pdf

I'd read that. They rigorously develop a theory for handling rank-n tensors, n > 2. This is pretty neat; I'll have to read it in more depth over christmas break.