  We study the eigenscheme of a matrix which encodes information about the
eigenvectors and generalized eigenvectors of a square matrix. The two main
results in this paper are this decomposition encodes the numeric data of the
Jordan canonical form of the matrix. We also describe how the eigenscheme can
be interpreted as the zero locus of a global section of the tangent bundle on
projective space. This interpretation allows one to see eigenvectors and
generalized eigenvectors of matrices from an alternative viewpoint.
