---
tag: 06DW9BAWB1
kind: object
name: Identity matrix
msc: [15A99]
instance_of: [0VD9ZSWWJG]
sources:
  - url: https://en.wikipedia.org/wiki/Identity_matrix
    locator: lead section
same_as:
  wikidata: Q193794
  wikipedia: Identity matrix
---

## Description

Let $R$ be a ring and let $n \ge 0$ be an integer.
The *identity matrix* $I_n$ is the $n \times n$ [[matrix|0VD9ZSWWJG]] over $R$ with entries $1$ on the main diagonal and $0$ elsewhere.
Its entries are $(I_n)_{ij} = \delta_{ij}$, where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ otherwise.
For each $m \times n$ matrix $A$, $A I_n = A$ and $I_m A = A$.

## Remarks

The identity matrix $I_n$ is the identity element of the [[matrix ring|0FHVPRAE60]] $M_n(R)$.
It is the matrix of the identity map in any basis.
For example, $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

## Explanations

### L1 human

The identity matrix has ones on the diagonal and zeros everywhere else.
It plays the role of the number $1$: multiplying by it changes nothing.
As a map, it leaves each vector where it is.