Sara Samy - Blog

Gershgorin and his circles

Fri, 25 Apr 2025 00:00:00 +0000

A theorem I love from linear algebra is the Gershgorin circle theorem which relates the spectrum of a $n \times n$ complex matrix to $n$ disks in the complex plane. Given $A \in \mathbb{C}^{n \times n}$ with spectrum A complex number $\lambda \in \mathbb{C}$ is an eigenvalue of $A$ if and only if there exists $v \in \mathbb{C}^{n}$ with $v \neq 0$ such that $$ Av = \lambda v $$or equivalently, $(A - \lambda I_{n})v = 0$. This implies that $A - \lambda I_{n}$ is a singular matrix. $$ \sigma(A) = \{ \lambda \in \mathbb{C} \ : \ \det(A - \lambda I_{n}) = 0 \} $$ we define $r_{i}$ to be the absolute row sum of the $i$-th row of $A$ with the diagonal entry $a_{ii}$ deleted, i.e. $$ r_{i} := \sum_{j \neq i} \lvert a_{ij} \rvert $$ The closed disk in the complex plane $$ D_{i} = \{ z \in \mathbb{C} \ : \ \lvert z-a_{ii} \rvert \leq r_{i} \} $$ with center $a_{ii}$ and radius $r_{i}$ is the $i$-th Gershgorin disk. The original result of Gershgorin states that the spectrum of $A$ is completely contained in the collection of all such disks. This collection is called the Gershgorin set which is closed and bounded in $\mathbb{C}$.

Solving the Bonnet problem - A hands-on adventure in 17 chapters

Thu, 15 Feb 2024 20:49:43 +0100

This page showcases the animations produced for the movie Solving the Bonnet problem (2023), directed by Ekaterina Eremenko. You can watch the trailer here.

The movie is based on the paper Compact Bonnet Pairs by Alexander I. Bobenko, Tim Hoffmann, and Andrew O. Sageman-Furnas.

Curvature lines of isothermic Torus

Let $f$ denote the parameterization function of the isothermic torus in the animations below. It has one family of planar curvature lines, i.e. the curve $u \mapsto f(u, v_{0})$ lies in a plane for every fixed $v_{0}$.

About

Mon, 01 Jan 0001 00:00:00 +0000

Mathematics student living in Berlin, Germany. 🦜🍃

sarrasamyy at icloud dot com

Github

Artworks (2009-2015)

Mon, 01 Jan 0001 00:00:00 +0000

Optimized textures (2015)

Group exhibition at Kestner Gesellschaft in Hannover, Germany.

3D-printed sculpture made on Zbrush.

Profit/Prophet (2014)

Group exhibition at St. Johannes-Evangelist-Kirche for Berlin Art Week 2014.

Was tust du, Objekt? (2015)

Group exhibition at Gypsum Gallery in Cairo.

Nile Sunset Annex (2014)

Documentation of a solo show at Nile Sunset Annex Gallery, an artist-run project in Cairo.