# Definition:Unit Matrix

## Definition

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\struct {\map {\MM_R} n, +, \times}$ be the ring of order $n$ square matrices over $R$.

Then the unit matrix (of order $n$) of $\struct {\map {\MM_R} n, +, \times}$ is defined as:

$\mathbf I_n := \sqbrk a_n: a_{i j} = \delta_{i j}$

where $\delta_{i j}$ is the Kronecker delta for $\map {\MM_R} n$.

That is, it is the square matrix where every element on the diagonal is equal to $1_R$, and whose other entries all are $0_R$.

## Examples

The $3 \times 3$ unit matrix is as follows:

$\mathbf I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$

## Also known as

Some sources call this the identity matrix, as it is the identity element of the ring of order $n$ square matrices over $R$.

There are several variants of $\mathbf I_n$ which can frequently be found, for example:

$\mathbf 1$
$\mathbf 1_n$
$\mathbb I_n$

In physics and mechanics it is common to see $\mathbf I$ used specifically to denote the $3 \times 3$ unit matrix.