Let and . Prove that and are similar.

Let and . Evidently, . In particular, and are similar.

(Computations performed with WolframAlpha; see here and here.)

unnecessary lemmas. very sloppy. handwriting needs improvement.

Compute the Jordan canonical form of the matrix . (Over .)

Let . Evidently, is in Jordan canonical form. (Computations performed with WolframAlpha; see here.)

Let . Find the Jordan canonical form for .

Clearly the characteristic polynomial of is , which factors completely over any field. Certainly does not satisfy , and a quick computation verifies that does not satisfy or . So the minimal polynomial of is , and thus the Jordan canonical form is .

Let . Compute the Jordan and rational canonical forms of over .

Let . Evidently, is in Jordan canonical form. Thus the elementary divisors of are , , and , so that the invariant factors of are and . So the rational canonical form of is . (Computations carried out by WolframAlpha; see here.)

Find all the possible Jordan canonical forms of matrices of dimension 2, 3, or 4 over .

We begin by finding the possible lists of invariant factors, starting with the possible minimal polynomials. Recall that every polynomial of degree at least 1 over has a root in , so that every polynomial is a product of linear factors.

If has dimension 2, then the characteristic polynomial of has degree 2 and thus the minimal polynomial has degree at most 2. The possible minimal polynomials are thus , , and . In this case, with the minimal polynomial chosen the remaining invariant factors are determined. So the possible lists of invariant factors are as follows.

- ,

The corresponding lists of elementary divisors are as follows.

- ,
- ,

And so the possible Jordan canonical forms are as follows.

Again, if has dimension 3, we can construct all the possible minimal polynomials, and in each case the remaining invariant factors are determined (in one case, without loss of generality). The possible lists of invariant factors are as follows.

- , ,
- ,
- ,

The possible lists of elementary divisors are as follows.

- , ,
- ,
- , ,
- ,
- , ,

The possible Jordan canonical forms are then as follows.

If has degree 4, evidently there are 11 possible minimal polynomials and 14 possible lists of invariant factors (in some cases, without loss of generality due to symmetry). The possible lists of invariant factors are as follows.

- , , ,
- ,
- , ,
- ,
- , ,
- ,
- ,
- ,
- ,

The possible lists of elementary divisors are as follows.

- , , ,
- ,
- , ,
- , , ,
- , , ,
- ,
- , ,
- , ,
- , , ,
- ,
- ,
- , ,
- , , ,

The corresponding Jordan canonical forms are as follows.

Let and . Show that these matrices have the same characteristic polynomial, but that one is diagonalizable and the other not. Compute the Jordan canonical form for each.

Let and . Evidently we have and , which are both in Jordan canonical form.

Recall that characteristic polynomials are invariant under conjugation; in particular, evidently both and have characteristic polynomial . By Corollary 24 in D&F, is not diagonalizable.

Prove that the matrices and are similar. Show that both can be diagonalized and give matrices and such that and are diagonal.

Let and . Evidently we have . In particular, and are similar and both are diagonalizable.

Determine which of the following matrices are similar: , , , and .

Okay, these matrix computations are getting to be excruciating to do by hand. I’m going to completely wimp out and use a CAS to find the matrices such that are in Jordan canonical form. The good news is that this does not render the proof incomplete, since we can easily verify that is in JCF. I wanted to get my hands dirty (so to speak) actually using the algorithm, but now that I’ve got a handle on it I’ll use a computer.

Note: WolframAlpha can compute the Jordan Canonical Form of a matrix with the syntax ‘jordan form A’ where A is a matrix in the form [[r11,r12,…,r1n],[r21,r22,…,r2n],[rn1,rn2,…,rnn]]. (Example.)

Let , , , and . Evidently then we have , , and . These products are in Jordan canonical form (obviously). So and are similar, and and are not similar to each other or to .

Compute the Jordan canonical form of .

We begin by computing the Smith normal form of . Evidently the following sequence of ERCOs achieves this.

The resulting matrix is . So the invariant factors of are and , and the elementary divisors are thus , , and . So the Jordan canonical form of is .