Test your intuition: is the following true or false? Assertion 1: If is a square matrix over a commutative ring , the rows of are linearly independent over if and only if the columns of are linearly independent over . (All rings in this post will be nonzero commutative rings with identity.) And how about […]| Matt Baker's Math Blog
I recently completed another summer internship at Meta (formerly Facebook). I was surprised to learn that one of the intern friends I met was an avid reader of my blog. Encouraged by the positive feedback from my intern friends, I decided to write another post before the end of summer. This post is dedicated to the mandem: Yassir, Amal, Ryan, Elvis, and Sam.| Jake Tae
In this post, we will take a look at Nyström approximation, a technique that I came across in Nyströmformer: A Nyström-based Algorithm for Approximating Self-Attention by Xiong et al. This is yet another interesting paper that seeks to make the self-attention algorithm more efficient down to linear runtime. While there are many intricacies to the Nyström method, the goal of this post is to provide a high level intuition of how the method can be used to approximate large matrices, and how ...| Jake Tae
Kernel and cokernel are dual concepts, but the former is far more likely to appear in textbooks. The cokernel is often in the background without a name.| John D. Cook
When I learnt about Cholesky decomposition it struck me immediately the elegance and simplicity of the solution. It was clear that the symmetry contained some sort of redundancy inside, but it was not clear at all how that symmetry could be exploited. Here I will try to pay my insignificant tribute to the man who discovered it and present you his story.| UnlinkedList
The dot product is a fundamental operation on two Euclidean vectors that captures a notion of similarity between the vectors. In this post, we’ll define the dot product and offer a number of angles for which to intuit the idea captured by this fundamental operation.| Matthew N. Bernstein
We will see that the 3D plot of \(x^2 + (y+zi)^2 = 1\) contains a circle and a hyperbola, where \(i\) is the imaginary number. Beyond the visuals, this helps us understand the (complex) eigenvalues of real matrices.| Toby Lam’s Blog
Coefficients of a product of polynomials can be generated by a multiplication between a matrix and a vector.| Toby Lam’s Blog
without eigendecomposition| Toby Lam’s Blog
Also why the JNF is ill-conditioned and how the QR decomposition works by considering projective transformations.| Toby Lam’s Blog
Throughout my blog posts on linear algebra, we have proven various properties about invertible matrices. In this post we bring, all of these statements into a single location and form a set of statements called the “invertible matrix theorem”. Each statement in the invertible matrix theorem proves that the matrix is invertible and implies all of the rest of the statements.| Matthew N. Bernstein
Singular components of a light transport matrix – for an explanation of what’s going on – keep on reading! In this post I’ll describe a small hike into the landscape of using line…| Bart Wronski
