Identity Matrix
The identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else, acting as the multiplicative identity for matrix multiplication.
Formula
I_n: \text{1s on diagonal, 0s elsewhere}
Definition
The identity matrix is a square matrix with $1$s running diagonally from top-left to bottom-right and $0$s everywhere else; multiplying any matrix by it gives back the original matrix, $AI = IA = A$, just as the number $1$ leaves ordinary multiplication unchanged. The $n \times n$ identity matrix $I_n$ has entries $(I_n)_{ij} = 1$ if $i = j$, $0$ otherwise, satisfies $AA^{-1} = A^{-1}A = I_n$ when an inverse exists, and has $\det(I_n) = 1$. It is the multiplicative identity of the ring $M_n(F)$, representing the identity linear map $\text{id}: F^n \to F^n$; in group theory $\{I_n\}$ is the trivial subgroup of $GL_n(F)$, its eigenvalues are all $1$, and it is the fixed point of every similarity transformation, $P^{-1}IP = I$ for any invertible $P$.
Example
The $2\times 2$ identity matrix $\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ leaves any matrix unchanged under multiplication, and $I_3 = \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}$ represents the "do nothing" transformation, multiplying a vector by $I_n$ returns the same vector. The identity matrix is the only matrix that is simultaneously symmetric, orthogonal, Hermitian, unitary, and diagonal, with trace $n$ and determinant $1$.
Key Insight
The identity matrix is to matrix multiplication what the number $1$ is to ordinary multiplication: it leaves everything unchanged, and it marks the boundary between matrices with positive determinant (orientation-preserving) and negative determinant (orientation-reversing), since $\det(I) = 1$. Just as $1$ generates the multiplicative group of units in $\mathbb{Z}/p\mathbb{Z}$, the identity is the starting point for constructing the general linear group $GL_n(F)$ and its rich structure of subgroups.