The Invariant Factor Decomposition Algorithm (Algebra 3: Lecture 4 Video 2)

Lecture 4: We started this lecture by considering three 3 x 3 matrices.  By computing the characteristic and minimal polynomial of each, we determined their invariant factors.  This allowed us to determine which pairs are similar to each other.  We noted that this idea of finding the invariant factors by computing the minimal and characteristic polynomial does not always work for larger matrices, so we showed how to find the invariant factors by diagonalizing the matrix xI-B over F[x] using elementary row and column operations.  As motivation for the rest of the lecture, we discussed some questions one might want to answer beyond 'Compute the Invariant Factors of this Matrix'.  We introduced the Invariant Factor Decomposition Algorithm and discussed how it led to a procedure for how to find the rational canonical form of an n x n matrix.  In the last part of the lecture we solved two example problems, first giving a description of all similarity classes of matrices over Q with a given characteristic polynomial, and then giving a description of the similarity classes of 3 x 3 matrices A with entries in Q satisfying A^3 = I.

Reading: We started this lecture by considering Example (1) on pages 482-483 of Section 12.2.  We then showed one way to diagonalize xI-B, which is the analogue of the computation for xI-A that is done on pages 483-484.  We then came back to earlier in Section 12.2 to give a description of the Invariant Factor Decomposition Algorithm.  We did not give all the details-- you can read them on pages 480-481.  We talked about how this leads to an algorithm to find the rational canonical form of a matrix.  Again, we did not give the details, but you can read them on pages 481-482.  At the end of the lecture we discussed Examples (4) and (5) at the end of Section 12.2.
I highly recommend that you carefully work through the numerical examples presented in Section 12.2 and make sure you understand the steps of the algorithms.