- #1
Chris L T521
Gold Member
MHB
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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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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.
