The [[Identity Element]] of [[Matrix Product|Matrix Multiplication]]. For any [[Matrix]] $A \in M_{m\times n}$, the identity value is a [[Matrix|Matrix]] that when multiplied by $A$ leaves no effect. $ \huge \begin{align} \forall A &\in M_{m\times m} \\ A I_{m} &= A \end{align} $ There is no identity matrix for any matrix with *non square dimensions*. For any matrix $A\in \R^n$, the [[Identity Matrix]] is equal to the zero matrix of dimensions $n\times n$ with all it's diagonal filled with $1$. $\huge \begin{align} I_{n} &= \underbrace{ \mat{ 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots &\vdots &\ddots &0 \\ 0 & 0 & \cdots &1 }}_{n} \end{align}$ >[!seealso] Transpose Property $\huge\transpose{I} = I$ >[!example] $\R^3$ >$\huge \begin{align} >I &\in \R^3 \\ >I &= \mat{1&0&0\\0&1&0\\0&0&1}\\ >\end{align} >$