Definition of orthogonal matrix

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.