Practical and Impractical Uses of Matrices
For many students like me, matrices are first introduced as nothing more than grids of numbers that we manipulate through various elementary operations. At first, it may feel like these calculations only exist for solving exam questions. However, once we explore how matrices work in real-life production systems, they become far more meaningful. Matrices are used to model transformations, analyze data, simulate systems, and process digital information.
Markov Chains
Model dynamic systems using Markov matrices. Predict future states and find equilibrium in systems like population dynamics.
Image Processing
Apply convolution kernels to transform images. Used in edge detection and was key in solving the Reginald Denny case.
Network Analysis
Represent social networks using adjacency matrices. Discover mutual friends and hidden patterns in connected data.
Based on "The Applications of Matrices"
This interactive page demonstrates several practical applications of matrices based on the video "What I wish my teachers told me way earlier." You can use the navigation menu to explore each application.
Markov Matrices: Modeling a Zombie Outbreak
This example demonstrates how Markov matrices can model dynamic systems. By assigning fixed probabilities (a 20% infection rate and a 10% cure rate), we can predict future human and zombie populations. Repeated multiplication by the Markov matrix leads to an equilibrium state because the system aligns with the eigenvector associated with eigenvalue 1.
Transition Probabilities (Per Hour)
Chance a human gets infected each hour
Chance a zombie gets cured each hour
Long-term Equilibrium
Eventually, the system settles at a point where the input equals the output. This happens because of the eigenvector with eigenvalue 1.
Population Projection (20 Hours)
Image Processing Through Convolution
Images are represented as matrices of pixel brightness values. When you apply a smaller matrix called a kernel through convolution, you can create effects like blurring, sharpening, and edge detection. The video mentions the Reginald Denny case, where edge detection revealed a tattoo in low-resolution footage. This shows how matrix operations actually contribute to real investigations.
Source Image Matrix
Grayscale 0-255Click pixels to draw. A simplified image with a vertical edge.
Apply Convolution Kernel
Processed Output
Network Theory and Adjacency Matrices
You can represent social networks using adjacency matrices, where a "1" means two people are connected. When you square the matrix, it reveals mutual friends between people. Cubing it shows triangular groups. This basically transforms social interaction into analyzable mathematical structures that get used in data science, sociology, and cybersecurity.
The Network (Graph)
6 People. Nodes represent individuals. Lines represent connections.
Adjacency Matrix (A)
| ID | 1 | 2 | 3 | 4 | 5 | 6 |
|---|
Matrix Powers Insight
When you calculate A² (the matrix squared), it shows you mutual friends