向量与矩阵的计算
【a】向量内积：dot
a⋅b=∑iaibi\rm \mathbf{a}\cdot\mathbf{b} = \sum_ia_ib_ia⋅b=∑iaibi
a = np.array([1,2,3]) b = np.array([1,3,5]) a.dot(b)
22
【b】向量范数和矩阵范数：np.linalg.norm
在矩阵范数的计算中，最重要的是ord参数，可选值如下：
| ord | norm for matrices | norm for vectors |
| None | Frobenius norm | 2-norm |
| 'fro' | Frobenius norm | / |
| 'nuc' | nuclear norm | / |
| inf | max(sum(abs(x), axis=1)) | max(abs(x)) |
| -inf | min(sum(abs(x), axis=1)) | min(abs(x)) |
| 0 | / | sum(x != 0) |
| 1 | max(sum(abs(x), axis=0)) | as below |
| -1 | min(sum(abs(x), axis=0)) | as below |
| 2 | 2-norm (largest sing. value) | as below |
| -2 | smallest singular value | as below |
| other | / | sum(abs(x)ord)(1./ord) |
matrix_target = np.arange(4).reshape(-1,2) matrix_target
array([[0, 1], [2, 3]])
np.linalg.norm(matrix_target, 'fro')
3.7416573867739413
np.linalg.norm(matrix_target, np.inf)
5.0
np.linalg.norm(matrix_target, 2)
3.702459173643833
vector_target = np.arange(4) vector_target
array([0, 1, 2, 3])
np.linalg.norm(vector_target, np.inf)
3.0
np.linalg.norm(vector_target, 2)
3.7416573867739413
np.linalg.norm(vector_target, 3)
3.3019272488946263
【c】矩阵乘法：@
[A∗m×pB∗p×n]∗ij=∑∗k=1pA∗ikB∗kj\rm [\mathbf{A} *{m\times p}\mathbf{B}* {p\times n}] *{ij} = \sum*{k=1}^p\mathbf{A} *{ik}\mathbf{B}* {kj}[A∗m×pB∗p×n]∗ij=∑∗k=1pA∗ikB∗kj
a = np.arange(4).reshape(-1,2) a
array([[0, 1], [2, 3]])
b = np.arange(-4,0).reshape(-1,2) b
array([[-4, -3], [-2, -1]])
a@b
array([[ -2, -1], [-14, -9]])
