Question

求线性变换在标准正交基下的矩阵设V是n维实内积空间,y 是V的单位向量,定义T:V→V,Tx=x-2(x,y)y,且已证

Answer 1

设 e1,e2,...,en 是V的标准正交基

设 y = k1e1+.+knen,

则 (ei,y) = ki

Te1 = e1-2(e1,y)y = e1 - 2k1 (k1e1+.+knen)

= (1-2k1^2)e1 -2k1k2e2 - ...-2k1knen

T(e1,e2,...,en) = (e1,e2,...,en) (按上写出矩阵A)

则 A = E - 2 (k1,k2,...,kn)(k1,k2,...,kn)^T

= E - 2yy^T