Linear Algebra in Python/v3

Learn how to perform several operations on matrices including inverse, eigenvalues, and determinents

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Imports¶

The tutorial below imports NumPy, Pandas, and SciPy.

In [1]:

import plotly.plotly as py
import plotly.graph_objs as go
from plotly.tools import FigureFactory as FF

import numpy as np
import pandas as pd
import scipy

Add Two Matrices¶

A Matrix is a 2D array that stores real or complex numbers. A Real Matrix is one such that all its elements $r$ belong to $\mathbb{R}$. Likewise, a Complex Matrix has entries $c$ in $\mathbb{C}$.

In [2]:

matrix1 = np.matrix(
    [[0, 4],
     [2, 0]]
)

matrix2 = np.matrix(
    [[-1, 2],
     [1, -2]]
)

matrix_sum = matrix1 + matrix2

colorscale = [[0, '#EAEFC4'], [1, '#9BDF46']]
font=['#000000', '#000000']

table = FF.create_annotated_heatmap(matrix_sum.tolist(), colorscale=colorscale, font_colors=font)
py.iplot(table, filename='matrix-sum')

Out[2]:

Multiply Two Matrices¶

How to find the product of two matrices

In [3]:

matrix1 = np.matrix(
    [[1, 4],
     [2, 0]]
)

matrix2 = np.matrix(
    [[-1, 2],
     [1, -2]]
)

matrix_prod = matrix1 * matrix2

colorscale = [[0, '#F1FFD9'], [1, '#8BDBF5']]
font=['#000000', '#000000']

table = FF.create_annotated_heatmap(matrix_prod.tolist(), colorscale=colorscale, font_colors=font)
py.iplot(table, filename='matrix-prod')

Out[3]:

Solve Matrix Equation¶

How to find the solution of $AX=B$

In [4]:

A = np.matrix(
    [[1, 4],
     [2, 0]]
)

B = np.matrix(
    [[-1, 2],
     [1, -2]]
)

X = np.linalg.solve(A, B)

colorscale = [[0, '#497285'], [1, '#DFEBED']]
font=['#000000', '#000000']

table = FF.create_annotated_heatmap(X.tolist(), colorscale=colorscale, font_colors=font)
py.iplot(table, filename='matrix-eq')

Out[4]:

Find the Determinant¶

In [5]:

matrix = np.matrix(
    [[1, 4],
     [2, 0]]
)

det = np.linalg.det(matrix)
det

Out[5]:

-7.9999999999999982

Find the Inverse¶

In [6]:

matrix = np.matrix(
    [[1, 4],
     [2, 0]]
)

inverse = np.linalg.inv(matrix)

colorscale = [[0, '#F1FAFB'], [1, '#A0E4F1']]
font=['#000000', '#000000']

table = FF.create_annotated_heatmap(inverse.tolist(), colorscale=colorscale, font_colors=font)
py.iplot(table, filename='inverse')

Out[6]:

Find Eigenvalues¶

In [7]:

matrix = np.matrix(
    [[1, 4],
     [2, 0]]
)

eigvals = np.linalg.eigvals(matrix)
print("The eignevalues are %f and %f") %(eigvals[0], eigvals[1])

The eignevalues are 3.372281 and -2.372281

Find SVD¶

How to find the Singular Value Decomposition of a matrix, i.e. break up a matrix into the product of three matrices: $U$, $\Sigma$, $V^*$

In [8]:

matrix = np.matrix(
    [[1, 4],
     [2, 0]]
)

svd = np.linalg.svd(matrix)

u = svd[0]
sigma = svd[1]
v = svd[2]

u = u.tolist()
sigma = sigma.tolist()
v = v.tolist()

colorscale = [[0, '#111111'],[1, '#222222']]
font=['#ffffff', '#ffffff']

matrix_prod = [
    ['$U$', '', '$\Sigma$', '$V^*$', ''],
    [u[0][0], u[0][1], sigma[0], v[0][0], v[0][1]],
    [u[1][0], u[1][1], sigma[1], v[1][0], v[1][1]]
]

table = FF.create_table(matrix_prod)
py.iplot(table, filename='svd')

Out[8]: