How do you find frobenius canonical form of a matrix?

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SUMMARY

The discussion focuses on finding the Frobenius canonical form of a matrix and establishing its similarity to a companion matrix. Participants emphasize that two matrices sharing the same Frobenius form are similar, and suggest that demonstrating both matrices have the same minimal polynomial is crucial. The conversation highlights the importance of eigenvalues and eigenvectors in this process, particularly in identifying a companion matrix with matching eigenvalues and eigenspace dimensions.

PREREQUISITES
  • Understanding of Frobenius canonical form
  • Knowledge of companion matrices
  • Familiarity with eigenvalues and eigenvectors
  • Concept of minimal polynomials
NEXT STEPS
  • Study the process of finding the Frobenius canonical form of a matrix
  • Learn how to construct and analyze companion matrices
  • Explore the relationship between eigenvalues, eigenvectors, and minimal polynomials
  • Investigate methods for proving matrix similarity
USEFUL FOR

Mathematicians, students studying linear algebra, and anyone interested in matrix theory and canonical forms.

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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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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?
 

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