Document Type
Article
Publication Date
Summer 7-30-2026
Abstract
What is a matrix? This is a question that many novice math students may come into a linear algebra class, thinking. Well, one definition is that they are a set of multiple linear equations, all smooshed into a simpler, condensed format meant to help identify multiple variables at once, or where all these variables intersect. Simpler matrices like 2 × 2 or 3 × 3 can be calculated by hand; however, when we get into larger sizes, say 1000 × 1000, things can get much more complicated. That is why the development of technology, such as digital computing, for solving them has made all the difference. Matrices are everywhere, from pixelated images on the average smartphone to MRI scans of the human brain. Their influence on our world today cannot be ignored. The real question is, how do they do this? In this paper, I am going to discuss a 'characteristic' that square matrices have in common, eigenvectors and eigenvalues. This brings us to the third question: What are eigenvalues? Well, they can be described as scalar values, but scalar values of what? A vector. What vector? An eigenvector, of course. Now, the next question: what is an eigenvector? An eigenvector is a special vector that does not change direction, only its magnitude (Blinn), A.K.A., its scalar, which is where the eigenvalues come in. These 'eigen' (which is just German for characteristic) are important in the context of technology and digital computing, as they are essential for stability, information storage, and data retrieval (Golub). While it is possible to calculate them via the famous 'characteristic' equation for the matrix, it can be nearly impossible for a larger square matrix. It would require a little help from computers. The rise in digital computing shifted our focus from analytical root equations to iterative approximations. Iterations are a repetition of calculations to get to a close or approximate solution, which is something computers are good at, especially with large matrices in a short period of time. This paper will discuss a few of these iterations, how they work, and what it means for the advancement of technology.
Recommended Citation
Onyenze, Adanna, "The Impact of Iterative Methods to Calculate Eigenvalues and Eigenvectors in Digital Computing" (2026). Faculty publications. 2.
https://digitalcommons.collin.edu/mathfaculty/2