---
tag: 63FAKMYZ2E
kind: concept
name: Square root of a matrix
msc: [15A16, 15A24]
sources:
  - url: https://en.wikipedia.org/wiki/Square_root_of_a_matrix
same_as:
  wikidata: Q964330
  wikipedia: Square root of a matrix
---

## Summary

A square root of a square matrix $A$ is a matrix $B$ with $B^2 = A$.
A matrix can have no square root, finitely many, or infinitely many.
A [[positive semidefinite matrix|01MMVW8CKC]] has exactly one positive semidefinite square root; see [[square root of a positive semidefinite matrix|0BJVDCF38Y]].
This square root is used in statistics, polar decompositions and quantum mechanics.

## Explanations

### L1 human

For numbers, a square root of $a$ is a number $b$ with $b^2 = a$.
For matrices the idea is the same, with matrix multiplication.
The diagonal matrix with entries $4$ and $9$ has the square root with entries $2$ and $3$, and also those with entries $\pm 2$ and $\pm 3$.
The identity matrix of size $2$ even has infinitely many square roots, such as every reflection of the plane.