Definition of identity matrix

Definition: The identity matrix $I_n$ (or simply $I$) is the $n \times n$ diagonal matrix with ones on the main diagonal and zeros elsewhere:

$$I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}$$

The identity matrix satisfies $AI = A$ and $IA = A$ for any compatible matrix $A$. It represents the identity transformation that maps every vector to itself.

Example:
$$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Properties:

• $I^{-1} = I$
• $\det(I) = 1$
• $\text{tr}(I) = n$
• $I^k = I$ for any positive integer $k$