Do you mean have no REAL eigenvalues? Or none at all?
Here is an example of a matrix with no real eigenvalues.
But when , we get an eigenvalue of .
The problem is that this 0 vector has no direction which contradicts the definition of an eigenvalue.
(ie. eigenvectors have the same direction when they undergo a transformation. They're only scaled by the eigenvalue. A 0 vector does not have a direction so cannot be an eigenvector).
for some scalar (there is usually no requirement that be non-zero here). So every non-zero vector is an eigen vector of the zero matrix, and is the corresponding eigenvalue.
So yes, I was wrong the zero matrix does have an eigen value (I had what had to be non-zero confused).