推 koos :強！ 03/02 00:11
推 anovachen :謝謝!! 03/03 14:30
(A^2)u = 0 => A(Au) = 0
i.e. Au lies in the range of A, donote by R(A),
and lies in the null space of A, donote by N(A).
And N(A) = R(A)^⊥ (if A is normal) ...(*)
=> Au also lies in R(A)^⊥
=> Au = 0. Q.E.D.
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Proof of (*):
Suppose u lies in R(A)^⊥, i.e. <u,Av> = 0 for all v in V.
Then <(A^*)u,v> = 0 for all v in V.
=> (A^*)u = 0
=> Au = 0, i.e. u lies in N(A).
Conversely, suppose u lies in N(A), i.e. Au = 0
=> (A^*)u = 0 and <(A^*)u,v> = <u,Av> = 0 for all v in V.
Hence, u lies in R(A)^⊥. Q.E.D.
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