Another example of using matrices is in the analysis of a flight network for
airlines. This will involve both airport hubs and smaller airports. In practice this
may involve hundreds of destinations – here we’ll look at a small example: the
hubs London (L) and Paris (P), and smaller airports Edinburgh (E), Bordeaux
(B), and Toulouse (T) and the network showing possible direct flights. To use a
computer to analyse such networks, they are first coded using matrices. If there
is a direct flight between airports a 1 is recorded at the intersection of the row
and column labelled by these airports (like from London to Edinburgh). The
‘connectivity’ matrix which describes the network above is A.
The lower submatrix (marked out by the dotted lines) shows there are no
direct links between the three smaller airports. The matrix product A × A = A^2 of
this matrix with itself can be interpreted as giving the number of possible
journeys between two airports with exactly one stopover. So, for example, there
are 3 possible roundtrips to Paris via other cities but no trips from London to
Edinburgh which involve stopovers. The number of routes which are either direct
or involve one stopover are the elements of the matrix A + A^2. This is another
example of the ability of matrices to capture the essence of a vast amount of data
under the umbrella of a single calculation.