Identity Matrix

The identity matrix is defined as the Matrix ${\hbox{\sf 1}}$ (or ${\hbox{\sf I}}$) such that

$${\hbox{\sf I}}({\bf X}) \equiv {\bf X}$$

for all Vectors ${\bf X}$. The identity matrix is

$$I_{ij} = \delta_{ij}$$

for $i,j = 1, 2$, ..., $n$, where $\delta_{ij}$ is the Kronecker Delta. Written explicitly,

$${\hbox{\sf I}} = \left[{\matrix{ 1 & 0 & \cdots & 0\cr 0 &... ...& \vdots & \ddots & \vdots\cr 0 & 0 & \cdots & 1\cr}}\right].$$