I have been studying Cooperstein's Advanced Linear Algebra for about seven months now and I am having problems understanding how to find the elementary divisors of a linear operator and how to find the invariant factors of a linear operator as well. I feel as though if I'm having trouble with these then I won't be able to move forward. I understand that the invariant factors comprise the characteristic polynomial and can find them using determinants and eigenvalues (I get that), but the text builds the theory up using direct sums, minimal polynomials of T, and T-cyclic subgroups and I'm just not seeing it.
As for an example,
Let $T$ be an element of $L(\mathbf R^4,\mathbf R^4)$ be the operator given by
$$
T(v) = \begin{pmatrix} -3&2&2&-4\\-3&1&4&-4\\-2&0&3&-2\\-1&0&2&-1\end{pmatrix}(v)
$$
Determine the elementary divisors and the invariant factors of $T$.
(how do I write the correct matrix in LaTeX?)
Best Answer
Here is an algorithm for finding the invariant factors using elementary methods. First find the minimal polynomial (the largest invariant factor). This can be done by finding the minimal polynomial of each vector in a basis and finding the least common multiple of of these polynomials. You can also find a maximal vector, v, whose minimal polynomial WRT the operator is the minimal polynomial of the operator. Set V' equal to the quotient of the space V by the T-invariant subspace generated by T, . Let T' be the operator on V' induced by T. Use the same algorithm to find the minimal polynomial of T'. This is the next invariant factor. You can continue in this way to find all the invariant factors without use of the Smith Form (which requires a knowledge of finite generated modules over Principal Ideal Domains (PIDs).
Finding the elementary divisors is problematic since it requires factoring the invariant factors into irreducible polynomials and there is no efficient algorithm for doing this.