Orthogonal Matrix

An orthogonal matrix is a real, square matrix which has all columns and rows being orthonormal vectors.

Definition §

Orthogonal Matrices ^formula

An orthogonal matrix is a real, square matrix, $\mathbf{Q}$ such that:

$${\mathbf{Q}}^{-1}={\mathbf{Q}}^T$$

This can be shown with:

$${\mathbf{Q}}^T\mathbf{Q}={\mathbf{I}}_n$$

Basis of Orthogonal Matrices ^theorem

A real $n\times n$ matrix is orthogonal iff it’s columns (or rows) form an orthonormal basis in the real vector space, ${\mathbb{R}}^n$, with the inner product being the dot product.

Properties §

Let $Q$ be an orthogonal matrix of size $n\times n$.

1. Orthogonal matrices preserve the dot product:

$$Q\vec{u}\cdot Q\vec{v}=\vec{u}\cdot \vec{v}\forall \vec{u},\vec{v}\in {\mathbb{R}}^n$$

A useful implication of this is that orthogonal matrices also preserve length and angles, since they are geometric properties defined by the inner product (in this case, the dot product).

2. The determinant of orthogonal matrices can only be 1 or -1

$$\det (\mathbf{Q})=\pm 1$$

Geometrically, orthogonal matrices represent rotations when $\det (\mathbf{Q})=1$, and reflections composed with rotations when $\det (\mathbf{Q})=-1$

Applications §

In Diagonalising Real Matrices §