设n为偶数,证明存在实数域上n阶方阵A,使A^2=-E.

题目

设n为偶数,证明存在实数域上n阶方阵A,使A^2=-E.

答案

首先,2阶方阵B = [0,1;-1,0]满足B^2 = [-1,0;0,-1] = -E.
对n = 2k,只需取A为分块对角矩阵diag(B,B,...,B),即可验证A^2 = -E.