Compute the Jordan canonical form of the matrix . Find an explicit matrix such that is in Jordan canonical form. Is diagonalizable? if not, why not?
We will follow the algorithm given on page 496 of D&F.
First we determine a sequence of ERCOs which transforms into Smith Normal Form. Evidently, the following sequence works.
The resulting matrix is . We used three row operations, whose corresponding column operations are as follows.
The resulting matrix is . Multiplying the only nonzero column successively by , , and to construct the columns of a matrix , we have . Evidently, , and is in Jordan canonical form.