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Exercise 10.B.3

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Proof. (a) The characteristic polynomial of $T$ is

(z - λ_{1}) (z - λ_{2}) \dots (z - λ_{n}) .

Expanding this product we see that the coefficient of $z^{n - 2}$ is the sum of the products of all pairs of distinct eigenvalues. In other words, it equals

λ_{1} \sum_{j = 2}^{n} λ_{j} + λ_{2} \sum_{j = 3}^{n} λ_{j} + \dots + λ_{n - 1} \sum_{j = n}^{n} λ_{j} .

(b) It equals the sum of the products of the eigenvalues of $T$ with one term missing:

\sum_{j = 1}^{n} \prod_{k \neq j} {(- 1)}^{n - 1} λ_{k} .

More succinctly, assuming $0$ is not an eigenvalue of $T$ :

\sum_{j = 1}^{n} {(- 1)}^{n - 1} \frac{\det T}{λ_{j}} .

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celiopassos

2017-10-06 00:00

Exercise 10.B.3

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