Identity Matrix
Functions & Advanced AlgebraThe 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 the identity matrix gives back the original matrix.
Example
The $2\times 2$ identity matrix: $\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$. Check: $\begin{bmatrix}2 & 3\\4 & 5\end{bmatrix} \times \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} = \begin{bmatrix}2 & 3\\4 & 5\end{bmatrix}$. Multiplication by $I_2$ does nothing.
Key Insight
The identity matrix is to matrix multiplication what the number $1$ is to ordinary multiplication: it leaves everything unchanged. $A \times I = I \times A = A$ for any square matrix $A$.
Definition
The $n \times n$ identity matrix $I_n$ has entries $(I_n)_{ij} = 1$ if $i = j$ (diagonal), $0$ otherwise. It satisfies $AI = IA = A$ for any $n \times n$ matrix $A$, and $AI = A$ for any $m \times n$ matrix $A$ (with appropriately sized $I$). $\det(I_n) = 1$.
Example
$I_3 = \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}$. The identity matrix represents the "do nothing" transformation: multiplying a vector by $I_n$ returns the same vector. $A A^{-1} = A^{-1} A = I_n$ (when the inverse exists).
Key Insight
The identity matrix is the boundary between matrices with positive and negative determinant: $\det(I) = 1$. Transformations with $\det > 0$ preserve orientation; $\det < 0$ reverses it. The identity preserves everything.
Definition
The identity matrix $I_n$ is the multiplicative identity of the ring $M_n(F)$, making it a unital ring. In terms of linear maps, $I_n$ represents the identity map $\text{id}: F^n \to F^n$. In group theory, $\{I_n\}$ is the trivial subgroup of $GL_n(F)$. The identity matrix has all eigenvalues equal to $1$ and is diagonalizable (trivially, it is already diagonal).
Example
The identity matrix is the only matrix that is simultaneously symmetric, orthogonal, Hermitian, unitary, diagonal, and has trace $n$ and determinant $1$. It is the fixed point of all similarity transformations: $P^{-1}IP = I$ for any invertible $P$.
Key Insight
The identity plays the role of "1" in the algebra $M_n(F)$. 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.