# Show companion matrix is similar to the following matrix

In summary, to show that the companion matrix is similar to the given matrix, one must show that the given matrix can be written as the product of an invertible matrix P, the companion matrix C, and the inverse of P. This can be done by reducing the problem to showing that P*A = C*P, and then finding the characteristic and minimal polynomials of the given matrix. If the minimal polynomial is the same as the characteristic polynomial, then the matrices are similar.

## 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.

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

oh lol fixed

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

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

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.

## 1. What is a companion matrix?

A companion matrix is a square matrix that is associated with a given polynomial. It is constructed by using the coefficients of the polynomial in a specific way.

## 2. How is a companion matrix related to a given matrix?

A companion matrix is said to be similar to a given matrix if they have the same eigenvalues and the same characteristic polynomial.

## 3. What is the significance of showing that a companion matrix is similar to a given matrix?

Showing that a companion matrix is similar to a given matrix can help in simplifying calculations and analyses of the given matrix, as the properties of the companion matrix can be used to make deductions about the given matrix.

## 4. How can you prove that a companion matrix is similar to a given matrix?

To prove that a companion matrix is similar to a given matrix, you can use the properties of similar matrices, such as having the same eigenvalues and characteristic polynomial, and perform row or column operations to show that they are equivalent.

## 5. Can a companion matrix be similar to more than one matrix?

Yes, a companion matrix can be similar to more than one matrix as long as they have the same eigenvalues and characteristic polynomial. This means that different matrices with the same eigenvalues can have companion matrices that are similar to each other.

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