# Show companion matrix is similar to the following matrix

1. Feb 20, 2014

1. The problem statement, all variables and given/known data
need to show companion matrix is similar to the following matrix
(here is the picture of the matrix)

2. Relevant equations

here is the companion matrix
http://en.wikipedia.org/wiki/Companion_matrix

information on matrix similarity
http://en.wikipedia.org/wiki/Matrix_similarity

3. The attempt at a solution

say given matrix is A and companion matrix is C then need to show

A = P^-1 * C * P for some invertible matrix P

i guess i could reduce guesswork by rewriting as

P*A = C*P

but even then it does not seem to be the ideal way to go about things.

Last edited: Feb 20, 2014
2. Feb 20, 2014

### kduna

I don't think you put in the right link to the matrix...

3. Feb 20, 2014

oh lol fixed

4. Feb 20, 2014

### kduna

What is the characteristic and minimal polynomial of that matrix? Can you construct the rational canonical form based off of elementary divisors?

5. Feb 20, 2014

i know char for companion but that is all.

i also know:

if same minimal poly then similar

if same frobenius canonical form then similar

but no clue how to go about finding them

6. Feb 20, 2014

### kduna

Since your matrix is upper triangular, finding the characteristic polynomial is easy: It is just $∏ (x-\lambda_i)$. If you can show that this is the minimal polynomial as well, then you are done.