This following is a very brief summary of what happened in class on 2024-11-19.
We first recapped the general description of the rational canonical form of a linear endomorphism of a finite-dimensional -vector space, . We then proceeded to outline an algorithm to compute the invariant factors of . The main work is to compute the Smith normal form, by starting with the matrix and then proceeding to use basic row and column operations to "diagonalize" this matrix into a matrix with a very specific shape.
We spent most of the class period working through an explicit example of computing the Smith normal form (and hence invariant factors) for a given matrix, . We noted how the invariant factors let us immediately write down the rational canonical form matrix, , for .
We then briefly mentioned how one can also obtain the change-of-basis matrix, , that will conjugate to . The process is a bit strange, but essentially involves first computing a helper matrix, , that encodes the -module generators for that establish the fundamental structural isomorphism. The matrix contains one nonzero column vector for each invariant factor (i.e., for each summand in the decomposition), which is an -generator for the corresponding -invariant subspace of . To get a basis for that summand, we repeatedly apply to that vector. See the notes linked above for the precise description.