The identity matrix of order $m$ is defined as the matrix $I_m\in\mathcal{M}_{m\times m}(\mathbb{K})$ having ones in the main diagonal and zeros elsewhere

$$ I_m = \left(\begin{array}{cccc} 1 & 0 & \cdots & 0\\ 0 & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1\\ \end{array}\right) $$

It may also be written as $I_m=(\delta_{ij})_{ij}$, where

$$ \delta_{ij}= \left\{ \begin{array}{rr} 1 & \text{if \(i=j\)}\\ 0 & \text{if \(i\neq j\)}\\ \end{array}\right. $$

is known as *Kronecker's delta*