# Identity matrix

In linear algebra, the square identity matrix is a $n \times n$ matrix with $1$s in its main diagonal and $0$s in every other entry. It is usually denoted $I_n$.

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

The corresponding linear map is the identity map. For any $n \times n$ matrix $A$, we have $AI_n = I_nA = A$. The inverse $A^{-1}$ of $A$ is the unique matrix such that $AA^{-1} = A^{-1}A = I_n$.

The determinant of $I_n$ is $1$. $I_n$ has only one eigenvalue $1$, occurring with multiplicity $n$. Hence, any $n \times n$ matrix is in the corresponding eigenspace. The characteristic polynomial of $I_n$ is $P_{I}(t) = (t-1)^n$, and the minimal polynomial is $t - 1$.