Friday, May 26, 2017

Applications: A list of projects using matrix operations

Matrix Operations

I've just finished teaching Linear Algebra twice since the beginning of the year, and I'll be teaching it again in the fall. It is time that I cleaned up my applications list and updated the project files. Since I am enumerating them, I might as well do it here. Here is part 1 on matrix operations.  Others may be added later.

Seriation in archaeology - Uses incident matrices, matrix multiplication and transposition, and the properties of symmetric matrices to determine relative ages of sites with common artifacts.  I haven't used this in my classes yet, but there seem to be some good resources available including "Some problems and method in statistical archaeology," David Kendall, World Archaeology, 1969, which is available in JStor.  It also appears in Gareth Williams Linear Algebra with Applications.

Color manipulation in images - Uses matrix multiplication to alter colors in the RGB scale.  This article by Paul Haeberli describes the 4x4 matrices needed to modify colors, including offsets.  I haven't used this in class yet, but I could see this as a good project to have students work in Mathematica.  It also is a companion for transformations in 3D graphics, which also use 4x4 matrices.  The ability to use matrix multiplication to add vectors is common to both areas.

Image color conversion - Uses matrix multiplication to convert from, say, RGB to YIQ, color models.  I haven't used this in class, but it shows up in Gareth Williams Linear Algebra with Applications. This article by Ford and Roberts describes a bevy of color models, and it seems that only some conversions are linear.  Without a way to test whether the color conversion is correct, I don't see this as an interesting project. However, maybe Mathematica can render the other color models.

Transformations in 2D graphics - Uses matrix multiplication to apply rigid and non-rigid transformations to images.  May or may not use projective coordinates, depending on whether translations are allowed.  Resources abound.

Projection of 3D images onto the plane - Uses matrix multiplication to project 3D images onto the plane given the coordinates of the image and the location of the viewer.  Uses projective coordinates.  I use a paper written by Jeanie Mullen, one of my honors students.  This project has worked best for students with programming backgrounds.

Two-port in an electrical circuit - Uses matrix multiplication to describe the change in voltage and current through a two-port or a series of two-ports.  A simple application of Ohm's law that creates two linear equations that can be described using matrix multiplication.  The equations relate the input current and voltage to the output current and voltage.  This Wikipedia article has a table of many transmission matrices and their effect.