Show companion matrix is similar to the following matrix

[catsarebad]

Homework Statement

need to show companion matrix is similar to the following matrix

(here is the picture of the matrix)

Homework Equations

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

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

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.

 

[kduna]

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

 

[catsarebad]

oh lol fixed

 

[kduna]

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

 

[catsarebad]

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

 

[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.