Game Development Reference
In-Depth Information
Wecanindexindividualelementsinamatrix. Thisisusuallydescribedasgettingthe(i,j)
th
element of the matrix, that is the element located at the i
th
row and j
th
column.
The
identity
matrix is a matrix that has the property that any matrix multiplied by the iden-
tity will be equal to the matrix.
Given a matrix
M
and the
identity matrix
I
:
The identity matrix consists of 1's down the diagonal starting at (0,0).
There are many useful operations that we will be using when working with matrices, per-
haps the most important one is
matrix multiplication
.
Matrix multiplication of linear transformations corresponds to the
composition
of linear
transformations. Given two linear transformations, the product of their multiplication is a
linear transformation that is the composition of the matrices.
To put it into context, we will often create the composition of three different transforma-
tions,
rotation, scale
and
position.
The multiplication of these matrices will yield a mat-
rix that will perform all three transformations in the order in which they were multiplied.
We will not delve too deeply into the proof or inner implementation of the different mat-
rix operations that we will use (e.g. matrix multiplication, inversion). The scope of imple-
menting these operations usually falls outside of the field of user interface programming.
Wewillfocusontheapplicabilityofthedifferentmatrixoperationsinthecontextofgraph-
ics and user interface development. That said, it is of great value to understand why these
operations work; you are highly encouraged to dig deeper and see for yourself what is go-
ing on underneath the hood.
There are excellent resources that cover matrix operation implementation in detail (James
M. Van Verth, 2004), (Tomas Akenine-Moller, 2008), (Gregory, 2009).
