🥥Python和Julia线性代数（矢量矩阵最小二乘）

矢量(向量)

向量是有序的有限数字列表。向量通常写成垂直数组，用方括号或弯括号括起来，如

$\left[\begin{array}{r} -1.1 \\ 0.0 \\ 3.6 \\ -7.2 \end{array}\right] \quad \text { 或 } \quad\left(\begin{array}{r} -1.1 \\ 0.0 \\ 3.6 \\ -7.2 \end{array}\right)$

它们也可以写成用逗号分隔并用括号括起来的数字。在这种符号风格中，上面的向量被写为

$(-1.1,0.0,3.6,-7.2)$

Python示例：

import numpy as np

x = np.array([-1.1,0.0,3.6,-7.2])
y = np.array([-1.1,0.0,3.6,-7.2])
x,y, len(x), len(y)

输出：

(array([-1.1,  0. ,  3.6, -7.2]), array([-1.1,  0. ,  3.6, -7.2]), 4, 4)

块或堆叠向量

x = np.array([1,-2]); y = np.array([1,1,0]);
z = np.concatenate((x,y))
z

输出：

array([ 1, -2,  1,  1,  0])

子向量和切片

x = np.array([9,4,3,0,5])
y = x[1:4]
y

输出：

array([4, 3, 0])

一阶差分向量

x = np.array([1,0,0,-2,2])
np.array([x[i+1]-x[i] for i in range(len(x)-1)])

输出：

array([-1,  0, -2,  4])

矩阵

矩阵是写在方括号之间的矩形数组，如

$\left[\begin{array}{cccc} 0 & 1 & -2.3 & 0.1 \\ 1.3 & 4 & -0.1 & 0 \\ 4.1 & -1 & 0 & 1.7 \end{array}\right]$

使用大圆括号代替方括号也很常见，如

$\left(\begin{array}{cccc} 0 & 1 & -2.3 & 0.1 \\ 1.3 & 4 & -0.1 & 0 \\ 4.1 & -1 & 0 & 1.7 \end{array}\right)$

Python示例

import numpy as np
import numpy.linalg as npl
import math
import matplotlib.pyplot as plt

pg112 = np.matrix([[0,1,1,0],[1,0,0,1],[0,0,0,1],[1,0,0,0]])
G = nx.from_numpy_matrix(pg112)
pos = nx.layout.spring_layout(G)
node_sizes = [3 + 10 * i for i in range(len(G))]
M = G.number_of_edges()
edge_colors = range(2, M + 2)
edge_alphas = [(5 + i) / (M + 4) for i in range(M)]

nodes = nx.draw_network_nodes(G, pos, node_size=node_sizes, node_color='blue', with_labels=True)
edges = nx.draw_network_edges(G, pos, node_size=node_sizes, arrowstyle='->',
                               arrowsize=10, edge_color=edge_colors,
                               edge_cmap=plt.cm.Blues, width=2)

ax = plt.gca()
ax.set_axis_off()
plt.show()

最小二乘法

Python和Julia处理线性代数

指点迷津 | Brief

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import numpy as np import numpy.linalg as npl import math import matplotlib.pyplot as plt pg112 = np.matrix([[0,1,1,0],[1,0,0,1],[0,0,0,1],[1,0,0,0]]) G = nx.from_numpy_matrix(pg112) pos = nx.layout.spring_layout(G) node_sizes = [3 + 10 * i for i in range(len(G))] M = G.number_of_edges() edge_colors = range(2, M + 2) edge_alphas = [(5 + i) / (M + 4) for i in range(M)] nodes = nx.draw_network_nodes(G, pos, node_size=node_sizes, node_color='blue', with_labels=True) edges = nx.draw_network_edges(G, pos, node_size=node_sizes, arrowstyle='->', arrowsize=10, edge_color=edge_colors, edge_cmap=plt.cm.Blues, width=2) ax = plt.gca() ax.set_axis_off() plt.show()