Can You Find the Minimal and Characteristic Polynomials of This Operator?

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In summary, a minimal polynomial is a unique monic polynomial of smallest degree that has the given operator as a root. It is used to find the characteristic polynomial. To find the minimal polynomial of an operator, one can use the Cayley-Hamilton Theorem or the characteristic polynomial, which involves setting the operator equal to its characteristic polynomial and solving for the minimal polynomial. A characteristic polynomial is a monic polynomial that describes the behavior of an operator on a vector space. It is obtained by setting the determinant of the operator's matrix representation equal to zero and solving for the polynomial. To determine the characteristic polynomial of an operator, one can find the matrix representation of the operator, take the determinant, and set it equal to zero. Finding the
  • #1
Chris L T521
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Here's this week's problem.

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Problem: Let $T: \mathbb{V}\rightarrow \mathbb{V}$ be an operator on a 4-dimensional real vector space $\mathbb{V}$. Assume that the characteristic polynomial of $T$ is $X^4-1$. Determine the minimal and characteristic polynomials for the operator $\bigwedge^2 T:\bigwedge^2\mathbb{V}\rightarrow \bigwedge^2\mathbb{V}$.

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Remark: Note that if $\mathbb{V}$ is an $n$-dimensional vector space, then $\displaystyle\dim\bigwedge\!\!\,^k\mathbb{V} = {n\choose k}$. It then follows that in our problem, the characteristic polynomial of $\bigwedge^2 T$ has degree 6.

 
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  • #2
No one answered this week's question.

As I have been busy as of late, I don't have a full solution right now to post -- I'll have one in the next 24 hours and will update this post, so stay tuned!
 

FAQ: Can You Find the Minimal and Characteristic Polynomials of This Operator?

What is a minimal polynomial?

A minimal polynomial is the monic polynomial of smallest degree that has the given operator as a root. It is unique and is used to find the characteristic polynomial.

How do you find the minimal polynomial of an operator?

To find the minimal polynomial of an operator, you can use the Cayley-Hamilton Theorem or the characteristic polynomial. The characteristic polynomial is usually the easier method and involves setting the operator equal to its characteristic polynomial and solving for the minimal polynomial.

What is a characteristic polynomial?

A characteristic polynomial is a monic polynomial that describes the behavior of an operator on a vector space. It is obtained by setting the determinant of the operator's matrix representation equal to zero and solving for the polynomial.

How do you determine the characteristic polynomial of an operator?

To determine the characteristic polynomial of an operator, you first need to find the matrix representation of the operator. Then, take the determinant of the matrix and set it equal to zero. The resulting polynomial will be the characteristic polynomial.

Why is it important to find the minimal and characteristic polynomials of an operator?

Finding the minimal and characteristic polynomials of an operator is important because they provide valuable information about the behavior of the operator on a vector space. The minimal polynomial tells us the smallest degree polynomial that has the operator as a root, while the characteristic polynomial helps us understand the eigenvalues and eigenvectors of the operator. This information can be useful in solving problems in linear algebra and other fields of mathematics.

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