标准正交向量:
[q_i^Tq_j =
egin{cases}
0, quad if ; i
eq j \
1, quad if ; i = j
end{cases}
]
令(Q = [q_1, q_2, dots ,q_n]),则(Q^TQ = I)
因此,当(Q)为方阵时,(Q^{-1} = Q^T),(Q)被称为正交矩阵。
假设(Q)拥有标准正交列,那么投影到它的列空间,投影矩阵为:
(P = Q(Q^TQ)^{-1}Q^T = QQ^T)
若(Q)为方阵,则(P = QQ^T = I)
考察方程(A^TAhat{x} = A^Tb),将(A)换成(Q),即:(Q^TQhat{x} = Q^Tb)
解得(hat{x} = Q^Tb),即(x_i = q_i^Tb)
格拉姆-施密特正交化:
(A = a)
(B = b - frac{A^Tb}{A^TA}A)
(C = c - frac{B^Tc}{B^TB}B - frac{A^Tc}{A^TA}A)
(dots)
标准化得:
(q_1 = frac{A}{lVert A
Vert}, q_2 = frac{B}{lVert B
Vert}, q_3 = frac{C}{lVert C
Vert} dots)
则:(Q = [q_1, q_2, q_3, dots])
(QR)分解:
(A = QR),其中(R)为一个上三角矩阵