Computer graphics

3D transformations

A 4×4 matrix transforms every vertex — translation, rotation, projection. Your GPU does billions of matrix multiplies per second.

Machine learning

weights of neural networks

A neural network layer is W·x + b — a matrix times a vector plus a bias. Training updates the matrix entries.

Solving linear equations

Gauss elimination

A·x = b solved by x = A⁻¹·b. From traffic-flow models to electrical circuits, this one equation rules them.

Cryptography

encoding messages

Hill cipher encrypts text by multiplying letter-vectors with an invertible matrix. Modern lattice-based crypto generalises this.

Search engines

PageRank uses huge matrices

PageRank is the dominant eigenvector of the web's link matrix. Google's original algorithm is a matrix equation.

Quantum mechanics

state representation

Quantum gates are 2×2 (or larger) unitary matrices. A quantum circuit is a chain of matrix multiplications.