- Thread starter
- Moderator
- #1
- Jan 26, 2012
- 995
Here's this week's problem.
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Problem: Let $V$ be a finite dimensional complex vector space. Let $\phi$ be an element of $\text{End}_{\mathbb{C}}(V)$, and consider the function $f:\mathbb{C}\rightarrow\mathbb{C}$ by \[f(z)=\det(1+z\cdot\phi).\]
Find an expression for $f^{\prime}(0)$ using what is known about trace, determinants, and the characteristic polynomial.
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Remember to read the POTW submission guidelines to find out how to submit your answers!
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Problem: Let $V$ be a finite dimensional complex vector space. Let $\phi$ be an element of $\text{End}_{\mathbb{C}}(V)$, and consider the function $f:\mathbb{C}\rightarrow\mathbb{C}$ by \[f(z)=\det(1+z\cdot\phi).\]
Find an expression for $f^{\prime}(0)$ using what is known about trace, determinants, and the characteristic polynomial.
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Remember to read the POTW submission guidelines to find out how to submit your answers!