Vector and matrix operations

Vector and matrix operations#

Teng-Jui Lin

Content adapted from UW AMATH 301, Beginning Scientific Computing, in Spring 2020.

The dot product of row vectors $u = [u_{1}, \dots, u_{n}]$ and $v = [v_{1}, \dots, v_{n}]$ is

u \cdot v = u v^{T} = \sum_{i = 1}^{n} u_{i} v_{i}

Problem Statement. Find the dot product of vectors

u = {[\begin{matrix} - 9 & 2 & 4 & 2 \end{matrix}]}^{T}, v = {[\begin{matrix} 4 & 3 & 1 & 0 \end{matrix}]}^{T}

import numpy as np
import scipy

u = np.array([-9, 2, 4, 2])
v = np.array([4, 3, 1, 0])

# dot product
u@v

-26

If $p \geq 1$ is a real number, then the $p$ -norm of vector $x = [x_{1}, \dots, x_{n}]$ is

‖ x ‖_{p} \equiv {(\sum_{i = 1}^{n} | x_{i} |^{p})}^{1 / p}

For example, when $p = 2$ , the 2-norm, or Euclidean norm, Frobenius norm, is

‖ x ‖_{2} \equiv {(\sum_{i = 1}^{n} | x_{i} |^{2})}^{1 / 2} = \sqrt{x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2}}

When $p = 1$ , the 1-norm, or Manhattan norm, is the sum of absolute value of the components

‖ x ‖_{1} \equiv {(\sum_{i = 1}^{n} | x_{i} |^{1})}^{1 / 1} = | x_{1} | + | x_{2} | + \dots + | x_{n} |

When $p \to \infty$ , the infinity norm, or maximum norm, is the maximum value of the absolute value of the component

‖ x ‖_{\infty} \equiv max_{i} | x_{i} |

Problem Statement. Find the 2-norm, 1-norm, and infinity norm of

u = {[\begin{matrix} - 9 & 2 & 4 & 2 \end{matrix}]}^{T}

# 2-norm, Frobenius norm
np.linalg.norm(u)

10.246950765959598

# 1-norm
np.linalg.norm(u, 1)

17.0

# infinity norm, maximum norm
np.linalg.norm(u, np.inf)

9.0

Problem Statement. Find the matrix product $AB$ where

\begin{array}{r} A = [\begin{array}{c} 1 & 4 & 5 & - 1 \\ 2 & - 3 & 1 & 3 \end{array}], B = [\begin{array}{c} 2 & 6 & 2 \\ 6 & 4 & 2 \\ 0 & 2 & - 1 \\ 1 & 5 & 6 \end{array}] \end{array}

A = np.array([[1, 4, 5, -1], [2, -3, 1, 3]])
B = np.array([[2, 6, 2], [6, 4, 2], [0, 2, -1], [1, 5, 6]])

# matrix multiplication
A@B

array([[ 25,  27,  -1],
       [-11,  17,  15]])