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 $n=200$ , $m=500$ .

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

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

Exercise 9.3: Norms
Compute the Frobenius norm of $A$ : $\Vert A\Vert_F$ and the infi nity 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）。

Exercise 9.4: Power iteration
Generate a matrix $Z$ , $n\times n$ , with Gaussian entries, and use the power iteration to fi nd 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。

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。

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。

Python-Numpy相关习题及解析

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