# What is an orthogonal matrix?

Oct 25, 2015

Essentially an orthogonal $n \times n$ matrix represents a combination of rotation and possible reflection about the origin in $n$ dimensional space.

It preserves distances between points.

#### Explanation:

An orthogonal matrix is one whose inverse is equal to its transpose.

A typical $2 \times 2$ orthogonal matrix would be:

${R}_{\theta} = \left(\begin{matrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{matrix}\right)$

for some $\theta \in \mathbb{R}$

The rows of an orthogonal matrix form an orthogonal set of unit vectors. For example, $\left(\cos \theta , \sin \theta\right)$ and $\left(- \sin \theta , \cos \theta\right)$ are orthogonal to one another and of length $1$. If we call the former vector $\vec{A}$ and the latter vector $\vec{B}$, then:

$\vec{A} \cdot \vec{B} = - \sin \theta \cos \theta + \sin \theta \cos \theta = 0$
(hence, orthogonal)

$| | \vec{A} | | = \sqrt{{\cos}^{2} \theta + {\sin}^{2} \theta} = 1$
$| | \vec{B} | | = \sqrt{{\left(- \sin \theta\right)}^{2} + {\cos}^{2} \theta} = 1$
(hence, unit vectors)

The columns also form an orthogonal set of unit vectors.

The determinant of an orthogonal matrix will always be $\pm 1$. If it is $+ 1$ then the matrix is called a special orthogonal matrix.