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施密特正交化
施密特正交化(Gram-Schmidt Orthogonality)是常用的求欧式空间正交基的方法。给定一个线性无关向量组 a 1 , a 2 , . . . , a m a_1,a_2,...,a_m a1,a2,...,am,经过施密特正交化可以得到一组正交向量组 b 1 , b 2 , . . . , b m b_1,b_2,...,b_m b1,b2,...,bm,正交向量组中的每个向量正交化之后,就得到一个标准正交向量组。 下面我们来看施密特正交化的具体步骤。因为向量组 a 1 , a 2 , . . . , a m a_1,a_2,...,a_m a1,a2,...,am是线性无关的,所以可以用这些向量来构造 b 1 , b 2 , . . . , b m b_1,b_2,...,b_m b1,b2,...,bm,具体如下: b 1 = a 1 b_1 = a_1 b1=a1 b 2 = a 2 + h a 1 b_2 = a_2 + ha_1 b2=a2+ha1 b 3 = a 3 + h 1 a 1 + h 2 a 2 b_3 = a_3 + h_1a_1 + h_2a_2 b3=a3+h1a1+h2a2 . . . ... ... 反过来,我们也可以从 b 1 , b 2 , . . . , b m b_1,b_2,...,b_m b1,b2,...,bm退回到 a 1 , a 2 , . . . , a m a_1,a_2,...,a_m a1,a2,...,am a 1 = b 1 a_1 = b_1 a1=b1 a 2 = b 2 + k b 1 a_2 = b_2 + k b_1 a2=b2+kb1 a 3 = b 3 + k 1 b 1 + k 2 b 2 a_3 = b_3 + k_1b_1 + k_2b_2 a3=b3+k1b1+k2b2 . . . ... ... = = ; ==; ==> b 1 = a 1 b_1 = a_1 b1=a1 b 2 = a 2 − k b 1 b_2 = a_2 - kb_1 b2=a2−kb1 b 3 = a 3 − k 1 b 1 − k 2 b 2 b_3 = a_3 - k_1b_1 - k_2b_2 b3=a3−k1b1−k2b2 . . . ... ... 只需要求出 k , k 1 , k 2 k,k_1,k_2 k,k1,k2这些系数,就可以得到 a 1 , a 2 , . . . , a m a_1,a_2,...,a_m a1,a2,...,am到 b 1 , b 2 , . . . , b m b_1,b_2,...,b_m b1,b2,...,bm的转化方式。由于 b 1 , b 2 , . . . , b m b_1,b_2,...,b_m b1,b2,...,bm是正交的,所以我们有: b 1 T b 2 = 0 ⇒ b 1 T ( a 2 − k b 1 ) = 0 ⇒ k = b 1 T a 2 b 1 T b 1 b_1^Tb_2=0 ⇒ b_1^T(a_2 - kb_1)=0 ⇒ k=\frac{b_1^Ta_2}{b_1^Tb_1} b1Tb2=0⇒b1T(a2−kb1)=0⇒k=b1Tb1b1Ta2 b 1 T b 3 = 0 ⇒ b 1 T ( a 3 − k 1 b 1 − k 2 b 2 ) = 0 ⇒ k 1 = b 1 T a 3 b 1 T b 1 b_1^Tb_3=0 ⇒ b_1^T(a_3 - k_1b_1 - k_2b_2)=0 ⇒ k_1=\frac{b_1^Ta_3}{b_1^Tb_1} b1Tb3=0⇒b1T(a3−k1b1−k2b2)=0⇒k1=b1Tb1b1Ta3 b 2 T b 3 = 0 ⇒ b 2 T ( a 3 − k 1 b 1 − k 2 b 2 ) = 0 ⇒ k 2 = b 2 T a 3 b 2 T b 2 b_2^Tb_3=0 ⇒ b_2^T(a_3 - k_1b_1 - k_2b_2)=0 ⇒ k_2=\frac{b_2^Ta_3}{b_2^Tb_2} b2Tb3=0⇒b2T(a3−k1b1−k2b2)=0⇒k2=b2Tb2b2Ta3 . . . ... ... 根据这个公式很容易写出施密特正交化代码: def schmidt_orthogonality(matrix_org, debug=False): """ b1 = a1, b2 = a2 - kb1, b3 = a3 - k1b1 - k2b2 :param matrix_org: m x n matrix, m >= n 且满秩 :return: """ m, n = matrix_org.shape matrix_ortho = matrix_org.copy() matrix_ortho = np.asarray(matrix_ortho, dtype=np.float) coefficient = np.zeros(shape=(m, n)) # 系数矩阵k、k1、k2 coefficient[0, 0] = 1 # b1 = a1 for i in range(1, n): # 开始处理下一列 coefficient[i, i] = 1 for j in range(i): b_j = matrix_ortho[:, j] k_j = np.dot(b_j, matrix_org[:, i]) / np.dot(b_j, b_j) coefficient[j, i] = k_j matrix_ortho[:, i] -= k_j * b_j # 正交向量b1,b2...做正交化处理,系数也做相应的改变 for i in range(n): devider = np.dot(matrix_ortho[:, i], matrix_ortho[:, i]) if abs(devider) |
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