**Definition:** A real square matrix $Q$ is **orthogonal** if its columns (and rows) form an orthonormal set:
$$Q^T Q = QQ^T = I \quad \text{or equivalently} \quad Q^{-1} = Q^T$$
This means the columns of $Q$ are mutually orthogonal unit vectors.
**Example:**
$$Q = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
is a rotation matrix (orthogonal with $\det = 1$).
**Properties:**
- $\det(Q) = \pm 1$
- $|\det(Q)| = 1$ (preserves volume)
- Preserves dot products: $(Qx) \cdot (Qy) = x \cdot y$
- Preserves lengths: $\|Qx\| = \|x\|$
- Preserves angles between vectors
- The product of orthogonal matrices is orthogonal
- The inverse of an orthogonal matrix is orthogonal
Orthogonal matrices represent **rotations** and **reflections** in Euclidean space.