identity matrix

Identity Matrix

The identity matrix $I$ is a square matrix where all diagonal entries are 1 and all off-diagonal entries are 0:

$$I=\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\end{array}\right]=({\delta }_{ij}{)}_{i,j=1}^{n,n}$$

where ${\delta }_{ij}$ is the Kronecker delta.

$AI=IA=A$ for any matrix $A$ of compatible dimensions, $(I{)}^n=I$, and $\det (I)=1$.

The eigenvalues of $I$ are all 1, since for any eigenvector $v$: $Iv=v$.