Linear Algebra Archive

Singular Value Decomposition Explained

In this post, we build an understanding of the singular value decomposition (SVD) to decompose a matrix into constituent parts. What is the Singular Value Decomposition? The singular value decomposition (SVD) is a way to decompose a

Linear Algebra for Machine Learning and Data Science

This series of blog posts aims to introduce and explain the most important mathematical concepts from linear algebra for machine learning. If you understand the contents of this series, you have all the linear algebra you’ll need

Eigenvalue Decomposition Explained

In this post, we learn how to decompose a matrix into its eigenvalues and eigenvectors. We also discuss the uses of the Eigendecomposition. The eigenvalue decomposition or eigendecomposition is the process of decomposing a matrix into its

Understanding Eigenvectors in 10 Minutes

In this post, we explain the concept of eigenvectors and eigenvalues by going through an example. What are Eigenvectors and Eigenvalues An eigenvector of a matrix A is a vector v that may change its length but

Gram Schmidt Process: A Brief Explanation

The Gram Schmidt process is used to transform a set of linearly independent vectors into a set of orthonormal vectors forming an orthonormal basis. It allows us to check whether vectors in a set are linearly independent.

Understanding the Change of Basis Matrix

In this post, we learn how to construct a transformation matrix and apply it to transform vectors into another vector space. This process is also referred to as performing a change of basis. As discussed in the

Orthogonal Matrix: Definition and Example

In this post, we introduce orthonormal bases, orthogonal matrices and discuss their properties. An orthogonal matrix is a square matrix whose rows and columns are vectors that are orthogonal to each other and of unit length. We

What is a Transpose Matrix

In this short post, we learn how to obtain the transpose of a matrix and how to perform operations with a matrix transpose. The transpose of a matrix is a matrix that is obtained by flipping the

Identity Matrix and Inverse Matrix

We introduce the inverse matrix and the identity matrix. In addition, we learn how to solve systems of linear equations using the inverse matrix. The identity matrix is a matrix in which the diagonal entries are 1,

How to Perform Matrix Multiplication

In this post we learn how to perform matrix multiplication and why we need to be mindful of matrix dimensions. Furthermore, we look at the properties of matrix multiplication. Matrix multiplication is an operation that consists of