Econometric Theory/Definitions

The identity matrix, with a size of n, is an n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is commonly denoted as ${\displaystyle I_{n}}$, or simply by I if the size is immaterial or can be easily determined by the context.

${\displaystyle I_{1}=[1]\quad I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\quad I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}}$

The most important property of the identity matrix is that, when multiplied by another matrix, A, the result will be A

${\displaystyle AI_{n}=A\,}$ and ${\displaystyle I_{n}A=A\,}$.