Graphics Reference
In-Depth Information
Chapter 4
Matrices
4.1 Introduction
Matrix notation
was investigated by the British mathematician, Arthur Cayley
(1821-1895), in 1858, fifteen years after the invention of quaternions. Cayley and
others had realised that it was possible to express a collection of equations by sepa-
rating constants and variables. For example, the following simultaneous equations
2
x
+
3
y
=
18
(4.1)
4
x
−
y
=
8
(4.2)
have a solution
x
=
4, which can be discovered by eliminating one vari-
able, such as
x
, and computing
y
, which in turn can be substituted into one of the
equations to reveal the value of
x
. However, matrix notation allows us to express the
equations as follows
3 and
y
=
23
4
x
y
18
8
=
(4.3)
−
1
where the array of four numbers is a
matrix
and the other two columns are
vectors
.
When multiplying the matrix and the vector
T
together we must multiply the
respective terms of the top row of the matrix with the column vector to equal 18 and
create (
4.1
). Similarly, we must multiply the respective terms of the bottom row of
the matrix with the column vector to equal 8 and create (
4.2
).
Matrix notation also allows us to express these equations as
Av
[
]
xy
=
c
(4.4)
where
23
4
,
x
y
,
18
8
.
A
=
v
=
c
=
−
1
There happens to be a special matrix such that when it multiplies a vector it
results in no change - this matrix is called an
identity matrix
and has the form
10
01
.
=
I
