设行矩阵P=（1/2,0,……0,1/2）,矩阵A=E-P^TP,B=E+2P^TP,其中E为n阶单位矩阵,求AB

Question

设行矩阵P=（1/2,0,……0 ,1/2）,矩阵A=E-P^TP,B=E+2P^TP,其中E为n阶单位矩阵,求AB

Answer 1

AB = (E-P^TP)(E+2P^TP)
= E - P^TP + 2P^TP - 2P^T(PP^T)P
= E + P^TP - 2 * (1/2) P^TP
= E