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Recently I needed to solve the following problem: given a square matrix $\mathbf{A}$ find another square matrix $\mathbf{B}$ satisfying $\text{vec}(\mathbf{A})^T \text{vec}(\mathbf{B}) = 0$. This is the sense in which I use the word orthogonal ...

(2024-04-06)

History 2020-04: website started as a pandemic project after learning that GitHub offered free hosting, written in plain HTML, CSS, and Javascript (a great learning experience) 2022-10: website transitioned to Hugo, a massive boon to maintainability and flexibility 2023-01: bought and ...

(2024-01-15)

During my office hours today I was asked whether “an extra pair of parentheses” would do any harm to the execution of a Python program. In the case at hand the answer was “no”, but I suspected there might be actually be a limit on ...

(2023-10-10)

Here we are concerned with “non-negative” linear systems, that is, linear systems where $\mathbf{A}, \mathbf{b} \geq \mathbf{0}$ (elementwise). In particular, we give a sufficient condition for ...

(2023-09-29)

The pivoted LU factorization of square matrices is the key routine for the direct solution of linear systems. All square, invertible matrices have a pivoted LU factorization of the form $\mathbf{PA} = \mathbf{LU}$, ...

(2023-06-06)

The LLVM Project, which focuses on “modular and reusable compiler technologies” has had an enormous impact on the field of compilers, and by extension almost every kind of “applied” computer science. Though compilers (at the time of ...

(2022-12-22)

An auto race I recently attended also ran a 50/50 raffle, which is where a single winner receives 50% of the money collected from ticket sales. Just like any form of gambling, it’s almost certainly a bad idea on average, but I wasn’t able to figure out ...

(2022-12-19)

An often stressed property of the conjugate gradient method (CG) for solving linear systems is the monotonic decrease in the A-norm of the error. When CG is applied in practice the exact solution is unknown and the error cannot be computed or tracked, so ...

(2022-11-29)

Like all fields of mathematics, linear algebra has many prominent figures whose names are non-trivial to pronounce in English (the lingua franca of science and mathematics). This is further complicated by numerical linear algebra libraries which have inconsistent ...

(2022-11-17)

Eigenpairs of symmetric matrices are intimately related to optimization and critical points, with the eigenvectors being critical points of the Rayleigh quotient. In optimization settings, the type of critical point (minimum, maximum, ...

(2022-07-29)

Although the majority of successful algorithms for the symmetric tensor eigenvalue problem use optimization techniques directly, there are a few notable algorithms that do not appear to be based on optimization. Rather, they more closely ...

(2022-05-29)

After watching a short documentary on the state and progress of competition speed climbing, with times going ever lower and quotes like “we never though X time was possible, and now were not sure how low times will go” I was ...

(2021-06-05)

A review of traditional style pizzas from national chains (specifically those in the Urbana, IL area) including: Domino’s, Jet’s, Little Ceasars, Papa John’s, and Pizza Hut. Domino’s standard/traditional pizza (top left) has a ...

(2021-02-15)

A review of deep dish and pan style pizzas from national chains (specifically those in the Urbana, IL area) including: Domino’s, Jet’s, Little Ceasars, Papa John’s and Pizza Hut. Domino’s pan pizza (top left) is appropriately doughy with a ...

(2020-10-30)

Gauss famously (and possibly fictionally) computed the sum between 1 and 100 at a young age. His original logic relies on symmetry and geometrical thought, especially triangles. In this write up we provide an alternative derivation of ...

(2020-10-19)

An old reference sheet with simple formulae for length, area, volume as well as useful conversions. It’s called , and is meant to be folded up and kept in a wallet or purse. This is certainly from a time before internet ...

(2020-05-13)

Input a number to determine if it is prime. The maximum input value before truncation error is 9007199254740991 (~9e15, nine and fifteen zeros!), which is not prime but has a relatively large least factor. This is because JavaScript uses double-precision floating-point numbers ...

(2020-04-25)

A summer day in 2019 reading Complex Variables and Applications by Churchill inspired what I now realize is almost certainly a useless (but perhaps slightly amusing) result. Enjoy.

(2019-07-19)

It is said Freeman Dyson attempted to complete this task at the age of 4 or 5. In this write up we complete the calculation and simultaneously present the necessary basics of chemistry and measurement.

(2018-04-19)