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Exercise 5.4.15

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The question in the Warning is because the definition of $f (A)$ is not $\det (A - AI)$ . To prove the version for matrix, we may apply the theorem to the linear transformation $L_{A}$ . So we know that the characteristic polynomial of $L_{A}$ is the same as that of $A$ , say $f (t)$ . And we get the result $f (L_{A}) = T_{0}$ by the theorem. This means

f (A) v = f (L_{A}) (v) = 0

for all $v$ . So we know $f (A) = O$ .

jephianlin

2011-06-27 00:00

Exercise 5.4.15

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