Game Development Reference
Inverse and Identity
Just as we can multiply a scalar by 1 to no effect, there is an identity transforma-
tion that produces the original vector. This is represented by the matrix E ,which
is a square diagonal matrix, sized appropriately to perform the multiplication on
the vector and with all 1s on the diagonal. For example, the following will work
for vectors in
Intuitively, this makes sense. If we examine the columns, we will see they are just
e 1 , e 2 ,and e 3 , thereby transforming the basis vectors to themselves.
Note that the identity matrix is often represented in other texts as I .Weare
using E to distinguish it from the inertial tensor, as discussed below.
The equivalent to standard division is the inverse. The inverse reverses the
effect of a given transformation, as represented by the following:
x = A − 1 Ax .
However, just as we can't divide by 0, we can't always find an inverse for a
transformation. First, only transformations from an n -dimensional space to an
n -dimensional space have inverses. And of those, not all of them can be inverted.
For example, the transformation
( x )= 0 has no inverse.
Discussing how to invert matrices in a general manner is out of the scope of
this chapter; it is recommended that the reader see [Anton and Rorres 94], [Golub
and Van Loan 93], or [Press et al. 93] for more information.
1.4.5 Affine Transformations
An affine transformation on a point x performs the basic operation
z = Ax + y ,
where A and y are a matrix and vector, respectively, of the appropriate sizes to
perform the operation. We can also represent this as a matrix calculation:
In general, in physical simulations, we are concerned with two affine transfor-
mations: translation (changing position) and rotation (changing orientation). (See