Data Science

Linear Algebra Cheat Sheet for Machine Learning

All of the Linear Algebra Operations that You Need to Use
in NumPy for Machine Learning.

The Python numerical computation library called NumPy provides many linear algebra functions that may be useful as a machine learning practitioner.

In this tutorial, you will discover the key functions for working with vectors and matrices that you may find useful as a machine learning practitioner.

This is a cheat sheet and all examples are short and assume you are familiar with the operation being performed.

You may want to bookmark this page for future reference.

Linear Algebra Cheat Sheet for Machine Learning

Linear Algebra Cheat Sheet for Machine Learning
Photo by Christoph Landers, some rights reserved.


This tutorial is divided into 7 parts; they are:

  1. Arrays
  2. Vectors
  3. Matrices
  4. Types of Matrices
  5. Matrix Operations
  6. Matrix Factorization
  7. Statistics

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1. Arrays

There are many ways to create NumPy arrays.


from numpy import array
A = array([[1,2,3],[1,2,3],[1,2,3]])


from numpy import empty
A = empty([3,3])


from numpy import zeros
A = zeros([3,5])


from numpy import ones
A = ones([5, 5])

2. Vectors

A vector is a list or column of scalars.

Vector Addition

c = a + b

Vector Subtraction

c = a - b

Vector Multiplication

c = a * b

Vector Division

c = a / b

Vector Dot Product

c =

Vector-Scalar Multiplication

c = a * 2.2

Vector Norm

from numpy.linalg import norm
l2 = norm(v)

3. Matrices

A matrix is a two-dimensional array of scalars.

Matrix Addition

C = A + B

Matrix Subtraction

C = A - B

Matrix Multiplication (Hadamard Product)

C = A * B

Matrix Division

C = A / B

Matrix-Matrix Multiplication (Dot Product)

C =

Matrix-Vector Multiplication (Dot Product)

C =

Matrix-Scalar Multiplication

C =

4. Types of Matrices

Different types of matrices are often used as elements in broader calculations.

Triangle Matrix

# lower
from numpy import tril
lower = tril(M)
# upper
from numpy import triu
upper = triu(M)

Diagonal Matrix

from numpy import diag
d = diag(M)

Identity Matrix

from numpy import identity
I = identity(3)

5. Matrix Operations

Matrix operations are often used as elements in broader calculations.

Matrix Transpose

B = A.T

Matrix Inversion

from numpy.linalg import inv
B = inv(A)

Matrix Trace

from numpy import trace
B = trace(A)

Matrix Determinant

from numpy.linalg import det
B = det(A)

Matrix Rank

from numpy.linalg import matrix_rank
r = matrix_rank(A)

6. Matrix Factorization

Matrix factorization, or matrix decomposition, breaks a matrix down into its constituent parts to make other operations simpler and more numerically stable.

LU Decomposition

from scipy.linalg import lu
P, L, U = lu(A)

QR Decomposition

from numpy.linalg import qr
Q, R = qr(A, 'complete')


from numpy.linalg import eig
values, vectors = eig(A)

Singular-Value Decomposition

from scipy.linalg import svd
U, s, V = svd(A)

7. Statistics

Statistics summarize the contents of vectors or matrices and are often used as components in broader operations.


from numpy import mean
result = mean(v)


from numpy import var
result = var(v, ddof=1)

Standard Deviation

from numpy import std
result = std(v, ddof=1)

Covariance Matrix

from numpy import cov
sigma = cov(v1, v2)

Linear Least Squares

from numpy.linalg import lstsq
b = lstsq(X, y)

Further Reading

This section provides more resources on the topic if you are looking to go deeper.


Other Cheat Sheets


In this tutorial, you discovered the key functions for linear algebra that you may find useful as a machine learning practitioner.

Are there other key linear algebra functions that you use or know of?
Let me know in the comments below.

Do you have any questions?
Ask your questions in the comments below and I will do my best to answer.

The post Linear Algebra Cheat Sheet for Machine Learning appeared first on Machine Learning Mastery.

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