More on Matrices - 3D Math Primer for Graphics and Game Development - page 172

Definition of orthogonal
matrix

MM T = I .

M is orthogonal ⇐⇒ (6.7)

Recall from Section 6.2.2 that, by definition, a matrix times its inverse is
the identity matrix ( MM

−1 = I ). Thus, if a matrix is orthogonal, its
transpose and inverse are equal:

Equivalent definition of
orthogonal matrix

M T = M

−1 .

M is orthogonal ⇐⇒

This is extremely powerful information, because the inverse of a matrix
is often needed, and orthogonal matrices arise frequently in practice in
3D graphics. For example, as mentioned in Section 5.7.5, rotation and
reflection matrices are orthogonal. If we know that our matrix is orthogonal,
we can essentially avoid computing the inverse, which is a relatively costly
computation.

6.3.2 Orthogonal Matrices—Geometric Interpretation

Orthogonal matrices are interesting to us primarily because their inverse is
trivial to compute. But how do we know if a matrix is orthogonal in order
to exploit its structure?
In many cases, we may have information about the way the matrix was
constructed and therefore know a priori that the matrix contains only ro-
tation and/or reflection. This is a very common situation, and it's very
important to take advantage of this when using matrices to describe rota-
tion. We return to this topic in Section 8.2.1.
But what if we don't know anything in advance about the matrix? In
other words, how can we tell if an arbitrary matrix M is orthogonal? Let's
look at the 3 × 3 case, which is the most interesting one for our purposes.
The conclusions we draw in this section can be extended to matrices of any
dimension.
Let M be an orthogonal 3 × 3 matrix. Expanding the definition of
orthogonality given by Equation (6.7), we have

M T

M = I ,

2 3 2 3 2 3

m 11 m 12 m 13
m 21 m 22 m 23
m 31 m 32 m 33

m 11 m 21 m 31
m 12 m 22 m 32
m 13 m 23 m 33

1 0 0

4 5 4 5 4 5

= 0 1 0 .

0 0 1