推 HP0:謝謝你的解答！ 09/27 15:21
※ 引述《HP0 (cksh)》之銘言：
: Let T be a linear operator on R3 which is represented by the matrix
: [3 1 -1]
: [2 2 -1]
: [2 2 0]
: Find a diagonalizable operator D on R3 and a nilpotent operator N on R3
: such that T=D+N and ND=DN
: 請問該如何著手去找？？
[1 0 0]
Find the Jordan form J of T s.t. T = PJ(P^-1) = P[0 2 0]P^-1
[0 1 2]
[1 0 0] [1 0 0] [0 0 0]
Now separate [0 2 0] into [0 2 0] + [0 0 0] = A + B,
[0 1 2] [0 0 2] [0 1 0]
then T = PJ(P^-1) = P(A+B)P^-1 = PAP^-1 + PBP^-1
Let D = PAP^-1 and N = PBP^-1,
since A is a diagonal matrix, B is nilpotent and AB = BA,
it's not difficult to verify that not only both D and N satisfy
properties we want, also ND = DN.
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