The concept of E-eigenvector (or just eigenvector) of a tensor was introduced by L. Qi in 2007. This talk is about schemes associated with tensor eigenvectors (called eigenschemes for tensors). The focus of this talk is on eigenschemes for matrices. I will show that information about the Jordan canonical form of a matrix is encoded in the primary decomposition of the ideal of the eigenscheme for the matrix. If time permits, schemes associated with eigenvectors of more general tensors will be discussed.