线代，矩阵

Question

线代，矩阵谢谢！！！

Answer 1

A^T
=(E-2XX^T)^T
=E^T-2(XX^T)^T
=E-2(XX^T)^T
=E-2XX^T
=A
又AA^T
=AA
=(E-2XX^T)(E-2XX^T)
=E-4XX^T+4XX^TXX^T
=E-4XX^T+4X(X^TX)X^T
=E-4XX^T+4X(1)X^T
=E-4XX^T+4XX^T
=E
则A是正交矩阵，满足
A^T=A^(-1)
综上所述，A=A^T=A^(-1)