Linear Algebra - Cyclic Decomposition

[Leitmotif]

Linear Algebra - Cyclic Decomposition, Rational Canonical Form

Homework Statement

I am given a 5x5 real matrix A, and I am looking for an invertible matrix P so that $$P^{-1}AP$$ is in rational form.

Homework Equations

The Attempt at a Solution

I calculated the characteristic polynomial and minimal polynomial, and they are the same in this case. There is only 1 eigenvalue, and my minimal polynomial is of the form $$p(x)=p_1p_2p_3$$ where $$p_i$$ are relatively prime polynomials. I can compute companion matrices so I know what the rational form looks like, but I am stumped on how to obtain that matrix P. Any pointers are appreciated.

 

[mighty2000]

Hello! It looks like you are trying to find the rational canonical form of a given matrix A. This is a common problem in linear algebra and it involves decomposing a given matrix into a direct sum of cyclic subspaces. The rational canonical form is a block diagonal matrix where each block corresponds to a cyclic subspace.

To find the rational canonical form, you can follow these steps:

1. Find the characteristic polynomial of the given matrix A.
2. Find the minimal polynomial of A, which is the smallest polynomial that satisfies p(A) = 0.
3. Factor the minimal polynomial into irreducible polynomials.
4. For each irreducible polynomial p_i, construct a cyclic subspace by finding a vector v_i such that v_i, Av_i, A^2v_i, ..., A^{n_i-1}v_i span the cyclic subspace, where n_i is the degree of p_i.
5. Repeat this process for each irreducible polynomial, until you have found all the cyclic subspaces.
6. The direct sum of these cyclic subspaces will give you the rational canonical form of A.
7. To find the invertible matrix P, you can choose a basis for each cyclic subspace and construct a matrix with these basis vectors as columns. This matrix P will be the invertible matrix you are looking for.

I hope this helps! Let me know if you have any further questions.