Question

设列向量a=（1,2,2）,A为三阶正交矩阵,则长度||Aa||= ________

Answer 1

知识点:正交变换不改变向量的长度.

因为 A为正交矩阵,所以有 A^TA = AA^T = E.

所以 (Aa,Aa) = (Aa)^T (Aa) = a^T A^T A a = a^T ( A^T A ) a = a^T E a = a^T a = (a,a).

所以 ||Aa|| = ||a|| = 根号( 1^2 +2^2 + 2^2) = 根号9 = 3