Matrices - Rotation Transforms for Computer Graphics - page 50

and

⎡ ⎤

014
314
426

⎣ ⎦ =

S + Q = A .

4.16 Inverting a Pair of Matrices

In later chapters we form the products of two or more matrices, and in some cases
require to find their inverse. In anticipation of this requirement, let's compute the
inverse of a pair of matrices.
Given two transforms T and R , the product TR and its inverse ( TR ) − 1

must

equal the identity matrix I :

( TR )( TR ) − 1

=

I

and multiplying throughout by T − 1

we have

T − 1 TR ( TR ) − 1 T − 1 =

R ( TR ) − 1 T − 1 .

=

Multiplying throughout by R − 1

we have

R − 1 R ( TR ) − 1 R − 1 T − 1 =

( TR ) − 1 R − 1 T − 1 . =

Therefore, if T and R are invertible, then
( TR ) − 1 R − 1 T − 1 . =

Generalising this result to a triple product such as STR we can reason that
( STR ) − 1 R − 1 T − 1 S − 1 . =

4.17 Eigenvectors and Eigenvalues

Matrices represent linear transforms that scale, translate, shear, reflect or rotate
points, whilst leaving the origin untouched. For example, the following 2D trans-
form

41
14

x
y

x
y

=

transforms the points on four unit squares as shown in Fig. 4.1 where we see a
pronounced stretching in the first and third quadrants, and reduced stretching in the
second and fourth quadrants.
It should be clear from Fig. 4.1 that any point (k, k) is transformed to another
point ( 5 k, 5 k) , and that its mirror point (

− k, − k) is transformed to ( − 5 k, − 5 k) .