Python-Numpy相关习题及解析

Generate matrices A, with random Gaussian entries, B, a Toeplitz matrix, where $A\in\mathbb{R}^{n\times m}$ and $B\in\mathbb{R}^{m\times m}$, for $n=200$, $m=500$.

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Exercise 9.1: Matrix operations
Calculate $A+A$, $AA^T$, $A^TA$ and $AB$. Write a function that computes $A(B-\lambda I)$ for any $\lambda$.
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$r\times c$ 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$n\times n$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$Bx=b$.
Answer:

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

• numpy.random.random() 函数返回$[0.0,1.0)$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 $A$: $\Vert A\Vert_F$ and the infinity norm of $B$: $\Vert B\Vert_\infty$. Also find the largest and
smallest singular values of $B$.
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 $Z$, $n\times n$, with Gaussian entries, and use the power iteration to find the largest
eigenvalue and corresponding eigenvector of $Z$.
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$Z$ 的特征值，第二个是矩阵$Z$ 的特征向量。

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Exercise 9.5: Singular values
Generate an $n\times n$ matrix, denoted by $C$, where each entry is 1 with probability $p$ and 0 otherwise. Use
the linear algebra library of Scipy to compute the singular values of $C$.
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$n\times m$ 的矩阵，其中每个数值是二项分布$\binom{n}{N}p^k(1-p)^{1-k}$ 中$N$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$n$ 和$p$ 的二项分布。
• numpy.linalg.svd(A) 函数对矩阵$A$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 $z$ and an array $A$ and finds the element in $A$ that is closest to $z$. 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$A$ 的绝对。
  • A.argmin() 函数返回矩阵$A$ 的最小值的下标。