?

?? 例1：

import numpy as npA = np.array([[2, 3], [3, -6]])w1, V1 = np.linalg.eig(A) # 計算A的特征值和特征向量print("A的特征值： = ", w1)print("A的特征向量： = ", V1)B = np.array([[5,2,0], [2,5,0], [-3,4,6]])w2, V2 = np.linalg.eig(B) # 計算B的特征值和特征向量print("/n");print("B的特征值 = ", w2)print("B的特征向量 = ", V2)

? 運行結果：

A的特征值： = ?[ 3. -7.]
A的特征向量： = ?[[ 0.9486833 ?-0.31622777]
?[ 0.31622777 ?0.9486833 ]]

B的特征值 = ?[6. 7. 3.]
B的特征向量 = ?[[ 0. ? ? ? ? ?0.57735027 ?0.36650833]
?[ 0. ? ? ? ? ?0.57735027 -0.36650833]
?[ 1. ? ? ? ? ?0.57735027 ?0.85518611]]

? 例2：

url：https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data

import numpy as npimport matplotlib.pyplot as pltfrom sklearn.decomposition import PCAimport pandas as pdfrom sklearn.preprocessing import StandardScaler# iris 數據的 URLurl = "xxx"# Pandas DataFramedf = pd.read_csv(url, names=["sepal length","sepal width","petal length","petal width","target"])nrow, ncol = df.shapeprint("Iris data set :", nrow, "records with", ncol, "attributes/n")print("First 5 records in iris data/n", df.head(5))features = ["sepal length", "sepal width", "petal length", "petal width"]x = df.loc[:, features].valuesy = df.loc[:,["target"]].valuesx = StandardScaler().fit_transform(x) # 變換為 平均0, 分散1 的數據pca = PCA(n_components=2) # 利用 PCAprincipalComponents = pca.fit_transform(x)# 利用2個主成分軸轉換為二維數據print("/nFirst principal axis:", pca.components_[0])print("Second principal axis:", pca.components_[1])principalDf = pd.DataFrame(data = principalComponents,              columns = ["principal component 1", "principal component 2"])finalDf = pd.concat([principalDf, df[["target"]]], axis = 1)print("/nFirst 5 Transformed records/n", finalDf.head(5))fig = plt.figure(figsize = (8,8))ax = fig.add_subplot(1,1,1)ax.set_xlabel("principal component 1", fontsize = 12)ax.set_ylabel("principal component 2", fontsize = 12)ax.set_title("PCA with 2 components", fontsize = 15)targets = ["Iris-setosa", "Iris-versicolor", "Iris-virginica"] # iris的數據類型名稱colors = ["r", "g", "b"] # 按類別指定的顏色for target, color in zip(targets,colors):    indicesToKeep = finalDf["target"] == target    ax.scatter(finalDf.loc[indicesToKeep, "principal component 1"]      , finalDf.loc[indicesToKeep, "principal component 2"], c = color, s = 40)ax.legend(targets)ax.grid()fig.show()

?? 運行結果：

--- A --- 1.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00 rank(A) = 4 --- B --- 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

rank(B) = 0 --- C --- 2.00 5.00 -3.00 -4.00 8.00 4.00 7.00 -4.00 -3.00 9.00 6.00 9.00 -5.00 2.00 4.00 0.00 -9.00 6.00 5.00 -6.00

rank(C) = 3 --- C^T --- 2.00 4.00 6.00 0.00 5.00 7.00 9.00 -9.00 -3.00 -4.00 -5.00 6.00 -4.00 -3.00 2.00 5.00 8.00 9.00 4.00 -6.00

rank(C^T) = 3

參考文獻

Introduction to Linear Algebra, International 4 th Edition by Gilbert Strang, Wellesley Cambridge Press.

百度百科[EB/OL]. []. https://baike.baidu.com/

本篇完。