Question

What is an orthogonal matrix?

Answer 1

Essentially an orthogonal `n xx 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 xx 2` orthogonal matrix would be:

`R_theta = ((cos theta, sin theta), (-sin theta, cos theta))`

for some `theta in RR`

The rows of an orthogonal matrix form an orthogonal set of unit vectors. For example, `(cos theta, sin theta)` and `(-sin theta, cos theta)` are orthogonal to one another and of length `1`. If we call the former vector `vecA` and the latter vector `vecB`, then:

`vecA cdot vecB = -sinthetacostheta + sinthetacostheta = 0`
(hence, orthogonal)

`||vecA|| = sqrt(cos^2theta + sin^2theta) = 1`
`||vecB|| = sqrt((-sintheta)^2 + cos^2theta) = 1`
(hence, unit vectors)

The columns also form an orthogonal set of unit vectors.

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