Application of Matrices
Following on from the last couple of posts on the subject, today we’re going to look at the application matrix operations and how they help us in the real world. Below I have constructed the diagram of a network which we can imagine has something flowing through it (like a railway system or water system etc.) and an “ideal system” has input nodes, branches, paths or “arcs” and output nodes.

In our diagram we have and
input nodes; and
,
and
output nodes. The numbers in each path represent the proportion of the trains or fluid or whatever is travelling through the network.
On looking at node we can see that the amount going to both
&
outputs is
so the amount going to
must be
. Likewise, in the case of
we can see that
gets
, and
gets
so
must get
.
Now that we know the nature of this system we can consider, if litres are input at
and
litres are input at
then the quantities
,
&
are the respective outputs at
,
&
respectively.
From this we can determine that…
…and we can express all of this information as an input-output equation:
// Note: the underlined letters are to indicate “who’s connected with who?” and “who ends up where?” in terms of data in the expression i.e. times
is
etc.
// Note: the proportions “flowing from a node” (either or
) are represented in the column of a matrix (thus they equal 1) and the proportions flowing into each node (
,
, &
) are written in a row i.e.
and
are proportions of whatever is flowing into
etc.
From the expression above, is the output vector and
is the input vector .
Working through an example:
The network from the previous example was expanded and the outputs ,
&
are now the inputs to for this new part of the system and those inputs now feed outputs
and
.

// We know that columns in a vector always add up to 1. In this instance, and
will be our column.
1) Find the relationship between the inputs ,
and the outputs
,
.
The “ relationship” between the inputs
,
&
and the outputs
&
is needed.
// Note: how we’re not looking straight away at the relationship between ,
,
&
(as I would naturally tend to try and do).
In this instance:
From the previous example:
By substituting out the ,
matrix we can say:
The first matrix is a by
and the other is a
by
, hence we can multiply and the result will be a
by
matrix.
This matrix gives us the relationship between inputs ,
and outputs
&