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I've been staring at this problem for a while now, and I just can't seem to figure it out. Any help is appreciated. I feel like I am overlooking something simple.

Let the characteristic equation of a linear transformation $T$ be $t^4 + 3t^2 + 5$. What is the trace of $T$?

Siong Thye Goh
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user469779
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1 Answers1

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The trace is the sum of characteristic roots, so if the characteristic polynomial $p(t)$ has leading term $t^n$, then the term of degree $n-1$ will have coefficient $-a$ where $a$ is the trace.

$$ p(t) = \prod_{i=1}^n (t-\lambda_i) = t^n - (\sum_{i=1}^n \lambda_i) t^{n-1} +\ldots $$

where the product and sum are both indexed by counting characteristic roots according to multiplicity.

hardmath
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