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Making developers awesome at machine learning
Making developers awesome at machine learning
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 manylinear 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.
Kick-start your project with my new bookLinear Algebra for Machine Learning, includingstep-by-step tutorials and thePython source code files for all examples.
Let’s get started.
Linear Algebra Cheat Sheet for Machine Learning
Photo byChristoph Landers, some rights reserved.
Overview
This tutorial is divided into 7 parts; they are:
- Arrays
- Vectors
- Matrices
- Types of Matrices
- Matrix Operations
- Matrix Factorization
- Statistics
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1. Arrays
There are many ways to create NumPy arrays.
Array
1 2 | fromnumpyimportarray A=array([[1,2,3],[1,2,3],[1,2,3]]) |
Empty
1 2 | fromnumpyimportempty A=empty([3,3]) |
Zeros
1 2 | fromnumpyimportzeros A=zeros([3,5]) |
Ones
1 2 | fromnumpyimportones A=ones([5,5]) |
2. Vectors
A vector is a list or column of scalars.
Vector Addition
1 | c=a+b |
Vector Subtraction
1 | c=a-b |
Vector Multiplication
1 | c=a *b |
Vector Division
1 | c=a/b |
Vector Dot Product
1 2 | c=a.dot(b) c=a@b |
Vector-Scalar Multiplication
1 | c=a *2.2 |
Vector Norm
1 2 | fromnumpy.linalgimportnorm l2=norm(v) |
3. Matrices
A matrix is a two-dimensional array of scalars.
Matrix Addition
1 | C=A+B |
Matrix Subtraction
1 | C=A-B |
Matrix Multiplication (Hadamard Product)
1 | C=A *B |
Matrix Division
1 | C=A/B |
Matrix-Matrix Multiplication (Dot Product)
1 2 | C=A.dot(B) C=A@B |
Matrix-Vector Multiplication (Dot Product)
1 2 | C=A.dot(b) C=A@b |
Matrix-Scalar Multiplication
1 2 | C=A.dot(2.2) C=A *2.2 |
4. Types of Matrices
Different types of matrices are often used as elements in broader calculations.
Triangle Matrix
1 2 3 4 5 6 | # lower fromnumpyimporttril lower=tril(M) # upper fromnumpyimporttriu upper=triu(M) |
Diagonal Matrix
1 2 | fromnumpyimportdiag d=diag(M) |
Identity Matrix
1 2 | fromnumpyimporteye I=eye(3) |
5. Matrix Operations
Matrix operations are often used as elements in broader calculations.
Matrix Transpose
1 | B=A.T |
Matrix Inversion
1 2 | fromnumpy.linalgimportinv B=inv(A) |
Matrix Trace
1 2 | fromnumpyimporttrace B=trace(A) |
Matrix Determinant
1 2 | fromnumpy.linalgimportdet B=det(A) |
Matrix Rank
1 2 | fromnumpy.linalgimportmatrix_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
1 2 | fromscipy.linalgimportlu P,L,U=lu(A) |
QR Decomposition
1 2 | fromnumpy.linalgimportqr Q,R=qr(A,'complete') |
Eigendecomposition
1 2 | fromnumpy.linalgimporteig values,vectors=eig(A) |
Singular-Value Decomposition
1 2 | fromscipy.linalgimportsvd U,s,V=svd(A) |
7. Statistics
Statistics summarize the contents of vectors or matrices and are often used as components in broader operations.
Mean
1 2 | fromnumpyimportmean result=mean(v) |
Variance
1 2 | fromnumpyimportvar result=var(v,ddof=1) |
Standard Deviation
1 2 | fromnumpyimportstd result=std(v,ddof=1) |
Covariance Matrix
1 2 | fromnumpyimportcov sigma=cov(v1,v2) |
Linear Least Squares
1 2 | fromnumpy.linalgimportlstsq b=lstsq(X,y) |
Further Reading
This section provides more resources on the topic if you are looking to go deeper.
NumPy API
Other Cheat Sheets
- Python For Data Science Cheat Sheet, DataCamp (PDF)
- Linear algebra explained in four pages (PDF)
- Linear Algebra Cheat Sheet
Summary
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.
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About Jason Brownlee
Jason Brownlee, PhD is a machine learning specialist who teaches developers how to get results with modern machine learning methods via hands-on tutorials.13 Responses toLinear Algebra Cheat Sheet for Machine Learning
Very helpful! thanks for compiling and sharing this cheat sheet.
Thanks for this quick cheat sheet. Very useful one!
Hello.
Thanks for this summary.
A small remark:
Matrix-Scalar Multiplication doesn’t work exclusively like that (at least in python 3.6)
you are able to use either A*scalar(k) or np.dot (k).
And it makes more sense to respect original math notations and not to abuse the function by using a scalar with it.
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