---
tag: J0N4J606X0
kind: concept
name: Hadamard matrix
msc: [05B20]
specializes: [0VD9ZSWWJG]
sources:
  - url: https://en.wikipedia.org/wiki/Hadamard_matrix
same_as:
  wikidata: Q1422682
  wikipedia: Hadamard matrix
---

## Summary

A Hadamard matrix is a square [[matrix|0VD9ZSWWJG]] with entries $\pm 1$ whose rows are pairwise orthogonal.
Their determinant has absolute value $n^{n/2}$, the largest possible for a real $n \times n$ matrix with entries in $[-1, 1]$.
Hadamard matrices give error-correcting codes, block designs and experimental designs.
The node [[Hadamard matrix|DM59GHSMAJ]] gives the precise definition.

## Explanations

### L1 human

Take a square grid of plus and minus signs.
It is a Hadamard matrix if any two rows agree in exactly half of the places and disagree in the other half.
The smallest interesting one is the $2 \times 2$ grid with rows $+\,+$ and $+\,-$.
Repeating this pattern in blocks gives Hadamard matrices of sizes $4, 8, 16$ and so on.