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Exercise 7.2.15

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The matrix

A = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{matrix})

has the characteristic polynomial to be $- t (t^{2} + 1)$ . Zero is the only eigenvalue of $T = L_{A}$ . But $T$ and $A$ is not nilpotent since $A^{3} = - A$ . By Exercise 7.2.13 and Exercise 7.2.14, a linear operator $T$ is not nilpotent if and only if the characteristic polynomial of $T$ is not of the form ${(- 1)}^{n} t^{n}$ .

jephianlin

2011-06-27 00:00

Exercise 7.2.15

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