Kronecker delta
Definition 1. We define the Kronecker delta $\delta_{ij}$ by $\delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j. \end{cases}$
Identity matrix
Definition 2. The $n \times n$ identity matrix $I_n$ is defined by $(I_n)_{ij} = \delta_{ij}$.
Remark
Remark. Let $A \in M_{n \times n}(F).$ Then $A$ is a diagonal matrix $\Longleftrightarrow A_{ij} = \delta_{ij} A_{ij}$ for all $i, j$.