Q2.5 For any matrix A 2 Rnn, if we use ai (i = 1, 2, : : : , n) to denote the ith column of the matrix A and use gi j = j cos i j j = jai aj j¹kai k kaj kº to denote the absolute cosine of the angle i j between any two vectors ai and aj (for all 1 i, j n), show that

where D is an n n matrix with its elements computed as di j = sign¹ai aj º¹kai kkaj kº ¹1 i, j nº, and B is an n n diagonal matrix with its diagonal elements computed as bii = ¹
Í
j=1 gi j ºkai k2 ¹1 i nº.

i=1 j=i+1 8ij) = (D-B)A,