This article explores the relationship between the eigenvalues and eigenvectors of AAᵀ and AᵀA for a real matrix A of any shape.
Explore the journey from the spectral properties of permutations to the geometry of doubly stochastic matrices, Birkhoff polytope, and how it all leads to Sinkhorn distances in optimization and machine learning.
This is a step-by-step implementation of quadratic and cubic Bezier curves from the ground up. Building upon the insight that polynomials are vectors too, it offers a linear algebriac view of this family of curves.
A common application of SVD is dimension reduction. It is used in an algorithm which allows to approximate n-dimensional points in a dataset to m-dimensional points where m < n.
This article visualizes three different interpretations of matrix products with the help of animations - i) weighted sums, ii) dot products and iii) outer products.
Kronecker product is a way to multiply two matrices of any shape. This apparently simple product has a lot of desirable properties and finds applications in different areas.
Starting from the idea of a small change in output given a small change in input, this series builds the concepts of linear operators, gradients and Jacobians, and covers the foundational ideas behind autodiff alongwith links to implementations in PyTorch and JAX.
Given a triangle on a 2d-plane with vertices [x1,y1], [x2,y2] and [x3,y3], and given a point [x,y], how can you find out whether the point is inside or outside the triangle?
Isometric projection is one of the ways to map a 3D point on a 2D plane. The matrix is derived using two different ways using first principles.