What is an orthogonal matrix?

1 Answer

Answer:

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.