Jordan Forms, Canonical Form

Canonical Form

A simple jordon block j_r(l) is an rxr matrix with l down the main diagonal and 1's on the subdiagonal.

A matrix is in jordan canonical form if it is block diagonal, and each block is a simple jordan block.

Any nilpotent matrix m is similar to a matrix in jordan canonical form. (This was proved in the previous section.) The jordan blocks all have zeros on the main diagonal, e.g. j_r(0). Thus the matrix m has zeros down its main diagonal. Being a lower triangular matrix, these are also the eigen values. This makes sense, for a single, nonzero eigen value would prevent m from being nilpotent. The corresponding eigen vector persists, scaled again and again by the eigen value.