# Matrix

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/matrix/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A rectangular grid of numbers representing a linear transformation. Term coined by Sylvester (1850); systematic theory by Cayley (1850s-60s).

## Key operations
Matrix multiplication (composition of transformations, non-commutative in general), identity matrix, inverse (AA⁻¹=I), invertibility ⟺ nonzero determinant.

## Special types
Symmetric (A=Aᵀ), Orthogonal (AᵀA=I), Diagonal, Hermitian (quantum observables)

## Historical origins
Nine Chapters on the Mathematical Art (China, c.200BCE-200CE) — array methods equivalent to Gaussian elimination. Sylvester (1850) coined "matrix." Cayley (1850s-60s) founded systematic matrix algebra.

## Eigenvalue decomposition & SVD
Eigenvectors/eigenvalues reveal invariant directions. Singular Value Decomposition (A=UΣVᵀ) works for any matrix — basis of recommendation algorithms, data compression.

## Computational complexity
Naive: O(n³). Strassen (1969): O(n^2.807). Active research continues (AlphaTensor, 2022).

## Sources
### Tier 1
- Strang, G. (2016). *Introduction to Linear Algebra*. 5th ed.
- Horn, R.A. and Johnson, C.R. (2012). *Matrix Analysis*. 2nd ed.
### Tier 2
- Strassen, V. (1969). *Numerische Mathematik*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*