How do you find frobenius canonical form of a matrix?

In summary, the conversation discusses how to show that a given matrix is similar to a companion matrix. The conversation mentions the possibility of using the Frobenius canonical form and determining the minimal polynomial of both matrices to prove their similarity. The question is also raised about finding the eigenvalues and eigenvectors of the given matrix and potentially using them to create a companion matrix.
  • #1
catsarebad
72
0
the actual problem is to show that

the given matrix is similar to companion matrix
ijn53.jpg


here is the companion matrix

Companion matrix - Wikipedia, the free encyclopedia

----------------

i know that if same frobenius canonical form then similar but i don't even know how to find the frobenius form. could someone point me in the right direcion?

also if i can show that both matrix have same minimal poly then i would be done but i don't see that being easy.
 
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  • #2
chickenwin said:
the actual problem is to show that

the given matrix is similar to companion matrix
ijn53.jpg


here is the companion matrix

Companion matrix - Wikipedia, the free encyclopedia

----------------

i know that if same frobenius canonical form then similar but i don't even know how to find the frobenius form. could someone point me in the right direcion?

also if i can show that both matrix have same minimal poly then i would be done but i don't see that being easy.

Hi chickenwin! Welcome to MHB! :)

Which eigenvalues and eigenvectors does your matrix have?
Can you find a companion matrix with the same eigenvalues and the same dimension of the eigenspace?
 

1. What is the Frobenius canonical form of a matrix?

The Frobenius canonical form of a matrix is a special form that a matrix can be transformed into through elementary row operations. It is a diagonal matrix with blocks of Jordan canonical form on the diagonal.

2. Why is the Frobenius canonical form important?

The Frobenius canonical form is important because it allows for a simpler representation of a matrix, making it easier to perform operations such as matrix multiplication and finding eigenvalues. It also reveals important information about the structure and properties of the matrix.

3. How do you find the Frobenius canonical form of a matrix?

To find the Frobenius canonical form of a matrix, you must first find the Jordan canonical form of the matrix. Then, using elementary row operations, you can transform the Jordan canonical form into the Frobenius canonical form by rearranging the blocks on the diagonal.

4. Can all matrices be transformed into the Frobenius canonical form?

No, not all matrices can be transformed into the Frobenius canonical form. Only square matrices with complex entries can be transformed into this form.

5. What are the benefits of using the Frobenius canonical form over other forms?

Compared to other canonical forms, the Frobenius canonical form is more computationally efficient in terms of finding eigenvalues and performing matrix operations. It also provides a clear representation of the structure and properties of the matrix, making it useful in applications such as solving systems of linear equations and studying linear transformations.

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