Similarity for n x n Matrices: n = 2,3,4 (Algebra 3: Lecture 3 Video 3)

Lecture 3: We started this lecture by proving two facts about characteristic polynomials.  We then proved that the characteristic polynomial of A is the product of the invariant factors of A.  As a consequence, we saw that the minimal polynomial divides the characteristic polynomial, and that the characteristic polynomial divides some power of the minimal polynomial.  We then solved several exercises from Section 12.2.  We first analyzed when 2 x 2 matrices are similar, and then analyzed when 3 x 3 matrices are similar.  We saw that two 4 x 4 matrices can have the same minimal polynomial and characteristic polynomial but have different invariant factors.  We then gave an interpretation of two of the coefficients of the characteristic polynomial of a matrix.  Finally, we discussed the idea of diagonalizing the matrix xI-A using elementary row and column operations, and stated that this process leads to the invariant factors of A.

Reading: In this lecture we closely followed the material of Section 12.2 pages 478-479.  We also solved Exercises 3,4, and 6 from Section 12.2.