设向量a为n维列向量,a^t*a=1,令H=E-2a*a^t,证明H是正交矩阵

H^TH = (E-2aa^t)^T(E-2aa^t)
= (E-2aa^t)(E-2aa^t)
= E-2aa^t-2aa^t+4aa^taa^t
= E-4aa^t + 4 a(a^ta)a^t
= E - 4aa^t + 4aa^t
= E
所以H是正交矩阵.