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

[Chris L T521]

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.

 

[Chris L T521]

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!