Jordan Canonical Form (Algebra 3: Lecture 5 Video 2)

Lecture 5: We started this lecture by recalling the Elementary Divisor Decomposition for a finitely generated module over a PID.  We recalled how to go from the Invariant Factor Decomposition to this Elementary Divisor Decomposition.  We saw what this gave us in the particular case where V is a finite dimensional vector space over F and T a linear transformation giving V the structure of an F[x]-module.  We focused on the special case where F contains all the eigenvalues of T.  We saw that this gave V as a direct sum of finitely many cyclic F[x]-modules of the form F[x]/((x-lambda)^k).  We saw how to choose a vector space of each of these quotients on which the 'multiplication by x' transformation had a particularly simple form.  This led to the definition of a Jordan block and what it means for a matrix to be in Jordan Canonical Form.  We saw that choosing bases for each cyclic factor in the Elementary Divisor Decomposition in this way, we got a basis of V such that the matrix of T with respect to this basis was in Jordan canonical form.  We saw that Jordan canonical form is unique up to permutation of the Jordan blocks.  We then discussed Jordan canonical form for matrices.  At the end of the lecture we discussed some connections between Jordan canonical form and diagonal matrices.  In particular, we proved that A is similar to a diagonal matrix if and only if the minimal polynomial of A has no repeated roots.

Reading: We started this lecture by recalling some material from Section 12.1 about Elementary Divisors.  If you do not remember this well, it might be helpful to look back over it.  We then followed the first part of Section 12.3, pages 491-494, pretty closely.  We also mentioned Exercise 2 of Section 12.2.