Show companion matrix is similar to the following matrix

  • Thread starter catsarebad
  • Start date
  • #1
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Homework Statement


need to show companion matrix is similar to the following matrix
F552C6O.jpg
(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.
 
Last edited:

Answers and Replies

  • #2
52
7
I don't think you put in the right link to the matrix...
 
  • #4
52
7
What is the characteristic and minimal polynomial of that matrix? Can you construct the rational canonical form based off of elementary divisors?
 
  • #5
60
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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
52
7
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.
 

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