Identity Matrix

Definition: Identity Matrix

The $n\times n$-identity matrix $I_n\in F^{n\times n}$over the field $F$ is the square matrix which has the multiplicative identity of $F$ as its entries on the diagonal and whose other entries are the additive identity of $F$:

$$I_n\stackrel{\text{def}}{=}\left[\begin{array}{ccc}{1}_F& \cdots & 0_F\\ ⋮& \ddots & ⋮\\ 0_F& \cdots & 1_F\end{array}\right]$$

Theorem: Multiplication with the Identity Matrix

The matrix product of any $m\times n$-matrix $A$ with the identity matrix $I_m$ on the left or the identity matrix $I_n$ on the right always results in the original matrix $A$.

$$I_{m}A=A{I}_n=A$$

PROOF

TODO