Python-Numpy相关习题及解析

Generate matrices A, with random Gaussian entries, B, a Toeplitz matrix, where A∈Rn×m A ∈ R n × m and B∈Rm×m B ∈ R m × m , for n=200 n = 200 , m=500 m = 500 .

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Exercise 9.1: Matrix operations
Calculate A+A A + A , AAT A A T , ATA A T A and AB A B . Write a function that computes A(B−λI) A ( B − λ I ) for any λ λ .
Answer:

import numpy as np

A = np.random.randn(200, 500)
B = np.random.randn(500, 500)

def fun(l):
    return np.dot(A, np.dot(B, l * np.eye(500)))

res1 = A + A
res2 = np.dot(A, A.T)
res3 = np.dot(A.T, A)
res4 = np.dot(A, B)
l = int(input("lambda: "))
res5 = fun(l)

• numpy.numpy.randn(r, c) 返回 r×c r × c 的标准正态（高斯）分布的矩阵
• numpy.dot(A, B) 函数计算矩阵A和矩阵B的积
• A.T 返回矩阵A的逆矩阵
• numpy.eye(n) 返回 n×n n × n 大小的单位矩阵

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Exercise 9.2: Solving a linear system
Generate a vector b with m entries and solve Bx=b B x = b .
Answer:

b = np.random.random((m,))
x = np.linalg.solve(B, b)

• numpy.random.random() 函数返回 [0.0,1.0) [ 0.0 , 1.0 ) 之间的随机数
• numpy.linalg.solve(A, b) 函数求解全秩线性矩阵方程Ax = b的解x

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Exercise 9.3: Norms
Compute the Frobenius norm of A A : ∥A∥F ‖ A ‖ F and the infi nity norm of B B : ∥B∥∞ ‖ B ‖ ∞ . Also find the largest and
smallest singular values of B B .
Answer:

res1 = np.linalg.norm(A, 'fro')
res2 = np.linalg.norm(B, np.inf)
res3 = np.linalg.norm(B, 2)
res4 = np.linalg.norm(B, -2)

• numpy.linalg.norm(x, ord=None) 函数的作用是：根据ord 参数的值，该函数能够返回八个不同矩阵范数中的一个其中,fro 返回弗罗贝纽斯（Frobenius）范数，numpy.inf 返回无限（infinity）矢范数，2 和 -2 分别返回最大和最小奇异值（largest and smallest singular values）。

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Exercise 9.4: Power iteration
Generate a matrix Z Z , n×n n × n , with Gaussian entries, and use the power iteration to fi nd the largest
eigenvalue and corresponding eigenvector of Z Z .
Answer:

Z = np.random.randn(n, n)
res1, res2 = np.linalg.eig(Z)

• numpy.linalg.eig(Z) 函数返回两个值，第一个是矩阵 Z Z 的特征值，第二个是矩阵 Z Z 的特征向量。

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Exercise 9.5: Singular values
Generate an n×n n × n matrix, denoted by C C , where each entry is 1 with probability p p and 0 otherwise. Use
the linear algebra library of Scipy to compute the singular values of C C .
Answer:

p = 0.7
n = 5
C = np.random.binomial(1, p, (n,n))
u, sigma, vt = np.linalg.svd(C)

• numpy.random.binomial(n, p, (n,m)) 函数返回一个 n×m n × m 的矩阵，其中每个数值是二项分布 (nN)pk(1−p)1−k ( n N ) p k ( 1 − p ) 1 − k 中 N N 的值，也就是可能成功的次数。或者说，得出的矩阵的数值的分布符合参数为 n n 和 p p 的二项分布。
• numpy.linalg.svd(A) 函数对矩阵 A A 进行奇异值分解，其中sigma 是奇异值数组。

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  Exercise 9.6: Nearest neighbor
  Write a function that takes a value z z and an array A A and finds the element in A A that is closest to z z . The
  function should return the closest value, not index.

  Hint: Use the built-in functionality of Numpy rather than writing code to find this value manually. In particular, use brackets and argmin.

  Answer:

  import numpy as np
  def find_nearest(array,value):
      idx = (np.abs(array-value)).argmin()
      return array[idx]
  
  value = 0.5
  
  A = np.random.random(10)
  print(find_nearest(A, value))

  • numpy.abs(A) 函数返回矩阵 A A 的绝对。
  • A.argmin() 函数返回矩阵 A A 的最小值的下标。