How to factorize a matrix in Julia

Overview

Linear algebra is a field of mathematics where we use linear representations in vector spaces. These representations involve matrices, equations, vectors, determinants, and their respective linear transformations.

We use the techniques and concepts of linear algebra in many fields, including computer engineering, data science and machine learning, economics, physics, geometry, and other disciplines that involve mathematical computations.

A matrix is one such important concept in linear algebra. Matrices consist of numbers arranged in rows and columns that represent transformations to ease computations.

Matrix factorizations in Julia

Since matrices are numbers arranged in rows and columns, their dimensions sometimes become large and complex.

To solve such complexities in matrices, we need factorization.

Factorization, also known as decomposition, is a technique we use to break down a matrix into smaller components or matrices to speed up computations and, sometimes, even save on memory.

There are different types of factorization techniques depending on the type of matrix. Julia can automatically apply the right kind of factorization technique depending on the matrix type or method specified.

Let’s discuss some matrix factorization types to understand which underlying techniques to apply and when.

Types of matrix factorizations

There are many types of factorization techniques applicable to matrices. For simplicity, let’s cover three:

1. The LU matrix factorization technique
2. The QR matrix factorization technique
3. The Cholesky matrix factorization technique

The LU matrix factorization technique

This is applicable to square (nxn) matrices. It decomposes a matrix A into L and U components.

A = LU

The QR matrix factorization technique

QR is applicable to non-square (mxn) matrices. It decomposes the matrix A into Q and R components.

A = QR

The Cholesky matrix factorization technique

This technique is applicable to the square but symmetric matrix. It decomposes the matrix A into into L, the lower triangular matrix, and L^T, the transpose of L.

This technique is especially useful in data science and machine learning, with applications in model optimization.

A = L.L^T

Practical applications in Julia

Some practical applications of matrices in Julia are as follows:

Using the factorize method

Julia automatically checks the best method to apply to a matrix, given its characteristics, and applies it when the factorize method is used.

using LinearAlgebra
A = randn(3,3)
print(A)
B = factorize(A)
print(B)

Using the MatrixFactorizations library

We use the MatrixFactorizations library for non-standard matrix factorization techniques, such as QL. It also currently implements other non-standard factorization techniques. These include RQ (a variant of QR), UL (a variant of LU), and a combined Cholesky factorization with inverse and polar decompositions.

using MatrixFactorizations
A = randn(3,3)
print(A)
print("--------------------------------------------")
B = ql(A)
print(B)

Code implementation in Julia

Explanation

• The first code uses the linear algebra library to factorize an mxn matrix. It automatically uses the QR technique to decompose the matrix.

Note: We can experiment with other nxn matrices to see which techniques are used.

• The second code uses the MatrixFactorization library to factorize an mxn matrix using the RQ technique. RQ is a variant of QR, with a difference in the order in which the matrices are decomposed.

Note: We can experiment with the UL and LQ techniques and see the results.