Orthogonal Matrix

TL;DR

A square real matrix whose transpose equals its inverse.
Multiplying by an orthogonal matrix preserves vector lengths, angles, and dot products.
Common examples include reflection matrices and rotation matrices in 2D.

Definition

An orthogonal matrix is a square matrix with real-valued entries that satisfies $A^T = A^{-1}.$ Equivalently, the rows of an orthogonal matrix are orthogonal to its columns, and vice versa.

Explanation

Two vectors x and y are orthogonal when their dot product is zero: $x \cdot y = 0.$ If such vectors appear as rows (or columns) of a square matrix, that matrix is orthogonal when the matrix’s transpose equals its inverse. Orthogonal matrices preserve geometric structure: when a vector is multiplied by an orthogonal matrix, its length (norm) and the angles between it and other vectors remain unchanged. They also preserve the dot product of two vectors: the dot product of the transformed vectors equals the dot product of the originals.

Examples

Example: reflection / sign matrix

The matrix $\begin{bmatrix}1 & 0 \\[4pt] 0 & -1\end{bmatrix}$ is its own transpose: $\begin{bmatrix}1 & 0 \\[4pt] 0 & -1\end{bmatrix}^T = \begin{bmatrix}1 & 0 \\[4pt] 0 & -1\end{bmatrix},$ so it is an orthogonal matrix.

Example: 2D rotation matrix

The rotation matrix by angle (\theta) is $\begin{bmatrix}\cos(\theta) & -\sin(\theta) \\[4pt] \sin(\theta) & \cos(\theta)\end{bmatrix}.$ Its transpose is $\begin{bmatrix}\cos(\theta) & \sin(\theta) \\[4pt] -\sin(\theta) & \cos(\theta)\end{bmatrix},$ which is the inverse of the original matrix, so the rotation matrix is orthogonal.

Use cases

Computer graphics (rotations and reflections that preserve shape and size)
Scientific and mathematical computations where preserving lengths, angles, or dot products is required

Transpose
Inverse
Dot product
Rotation matrix