- Find the minimum polynomial for
assuming the field of scalars is the rational numbers.

- Show, using the rational root theorem, the minimum polynomial for A in the above problem is irreducible with respect to ℚ. Letting the field of scalars be ℚ find the rational canonical form and a similarity transformation which will produce it.
- Letting the field of scalars be ℚ, find the rational canonical form for the matrix
- Let A : ℚ
^{3}→ ℚ^{3}be linear. Suppose the minimum polynomial is. Find the rational canonical form. Can you give generalizations of this rather simple problem to other situations? - Find the rational canonical form with respect to the field of scalars equal to ℚ for the
matrix
Observe that this particular matrix is already a companion matrix of λ

^{3}−λ^{2}+ λ− 1. Then find the rational canonical form if the field of scalars equals ℂ or ℚ + iℚ. - Suppose you have two n × n matrices A,B whose entries are in a field F and suppose G is an
extension of F. For example, you could have F = ℚ and G = ℂ. Suppose A and B are similar
with respect to the field G. Can it be concluded that they are similar with respect to
the field F? Hint: First show that the two have the same minimum polynomial over F.
Let this be ∏
_{i=1}^{q}ϕ_{i}^{pi}. Say β_{v j}is associated with the polynomial ϕ^{pj}. Thus, as described aboveequals p_{j}d. Consider the following table which comes from the A cyclic setIn the above, α

_{k}^{j}signifies the vectors below it in the k^{th}column. None of these vectors below the top row are equal to 0 because the degree of ϕ^{pj−1}λ^{d−1}is dp_{j}− 1, which is less than p_{j}d and the smallest degree of a nonzero polynomial sending v_{j}to 0 is p_{j}d. Thus the vector on the bottom of each column is an eigenvector for ϕ. Also, each of these vectors is in the span of β_{vj}and there are dp_{j}of them, just as there are dp_{j}vectors in β_{vj}. Now show thatis independent. If you string these together in opposite order, you get a basis which yields a block composed of sub-blocks for the Jordan canonical form for ϕ. Now show that ϕ,ϕare similar with respect to G. Therefore, they have exactly the same Jordan canonical form with respect to G. The lengths of the cycles are determined according to the size of these Jordan blocks and so you must have the same lengths of cycles for A as you do for B which shows they must have the same rational canonical form. Therefore, they are similar with respect to F. Thus all you have to do is to verify that the set is independent and that ϕ,ϕare similar with respect to G.To show why the set is independent, suppose

Then multiplying both sides by ϕ

^{pj−1}. Explain whyNow if any of the c

_{i0}is nonzero this would imply there exists a polynomial having degree smaller than p_{j}d which sends v_{j}to 0. This zeros out the c_{i0}scalars. Next multiply by ϕ^{pj−2}and give a similar argument. This kind of argument is used by Friedberg Insel and Spence [13] to verify uniqueness of the rational canonical form. Note that this is a very interesting result, being even better than the theorem which gives uniqueness of the rational canonical form with respect to a single field.

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