Definition:Unit Matrix

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

Let $\map {\mathcal M_R} n$ be the $n \times n$ matrix space over $R$.

Then the unit matrix (of order $n$) of the ring $\struct {\map {\mathcal M_R} n, +, \times}$ of $n \times n$ square matrices over $R$ is defined as:


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

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

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

Also known as
Some sources call this the identity matrix, as it is the identity element of the ring $\struct {\map {\mathcal M_R} n, +, \times}$ of $n \times n$ 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.

Also see

 * Unit Matrix is Unity of Ring of Square Matrices


 * Compare with, but do not confuse with, the Definition:Ones Matrix.