若α为n维列向量,则 A 应该是 A=E-αα^T.
证明:(1) A^T = (E-αα^T)^T
= E^T-(α^T)^Tα^T
= E-αα^T
= A.
所以A是对称矩阵.
(2) A^2 = (E-αα^T)^2 = E - 2αα^T + α(α^Tα)α^T = E-αα^T = A.
即A^2 = A
(3) 若A可逆
则由 A^=A 得 A = E
则 αα^T = 0
即有 α = 0.
这与 α^Tα = 1 矛盾.
所以 |A| = 0.