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# Definition of identity matrix

Put content here.**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:
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$$I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}$$
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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.
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**Example:**
$$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
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Properties:
- $I^{-1} = I$
- $\det(I) = 1$
- $\text{tr}(I) = n$
- $I^k = I$ for any positive integer $k$

# Parents

* Basic terminology and notation