Question

设方阵 A=E-2aaT,其中 E 为 n 阶单位矩阵,a 为 n 维单位列向量,证明：任意n维向量B都有//AB//=

Answer 1

分三步:

1.因为 a 为 n 维单位列向量,所以有

a'a = 1 ( 记 a' = aT )

2.A'A = (E-2aa')(E-2aa') = E - 4aa' + 4aa'aa' = E-4aa'+4aa' = E

3.||AB|| = √(AB)'(AB) = √B'A'AB = √B'B = ||B||.