## Eigenvectors of Tensors

Eigenvectors and eigenvalues of matrices are well understood: they are vectors and scalars that describe the magnitude and direction with which a matrix acts on some space. Tensors can be thought of as higher dimensional matrices, for example 3D arrays populated by numbers. For these, we can also define eigenvectors and this is a useful thing to do (for physics, and statistical models, etc.)

There is a simple algorithm to find an eigenvalue and eigenvector of a matrix: start at some random vector $v$ and repeatedly apply matrix $M$:

$v \to Mv \to M^2 v \to \cdots$

It’s easy to see that this method will always converge to the eigenvector corresponding to the largest eigenvalue. But for tensors the story is different: depending on the location of our starting vector we will converge to different eigenvectors. This picture described the situation for a random symmetric 3D tensor from a computer simulation. We label random starting vectors by the eigenvector that they eventually converged to – the colours represent the

“regions of convergence”

of the different eigenvectors.

The picture reveals an interesting structure about the tensor, and it also looks a bit like a sad fish!