---
tag: 1HS4RT7HK5
kind: concept
name: Random matrix
msc: [15B52, 60B20]
sources:
  - url: https://en.wikipedia.org/wiki/Random_matrix
same_as:
  wikidata: Q176604
  wikipedia: Random matrix
---

## Summary

A random matrix is a matrix whose entries are [[random variables|0PNRKMRC55]] on a common probability space.
Random matrix theory studies the laws of its [[eigenvalues|0BQAA2BK3Q]] and singular values, mostly when the size of the matrix grows.
For a large symmetric matrix with independent centred entries of equal variance, the scaled eigenvalues follow the [[Wigner semicircle distribution|3QTT95DMFH]].
Random matrices are used in nuclear physics, multivariate statistics, number theory and wireless communication.

## Explanations

### L1 human

Fill a square table with numbers drawn at random, for example by coin tosses or from a normal distribution.
The result is a random matrix, and its eigenvalues are random numbers too.
For one small matrix the eigenvalues look erratic.
For a large symmetric matrix with independent entries of mean $0$ and equal variance, a histogram of the eigenvalues, after scaling, has an almost fixed shape: a half circle.
This shows that randomness at the level of entries can create order at the level of the spectrum.