The Math of AHP -- Computing priorities from pairwise comparisons

**Eigenvector computation**

Priorities for the elements in a cluster are derived from the relative pairwise
comparisons

by normalizing the principle right eigenvector of the matrix consisting of the
pairwise

comparisons. An insight into why this is so 'natural’ a computation can be
obtained

by considering n rocks of known weights w1, w2,.. wn. We form a matrix of
the

ratio of the known weights of the rocks as follows:

If we wanted to "recover" or find the vector of weights, [w1, w2, w3, ... wn]
given

these ratios, we can take the matrix product of the matrix A with the vector w
to

obtain:

If we knew A, but not w, we could solve the above for w. The problem of solving

for a nonzero solution to this set of equations is very common in engineering
and

physics and is known as an eigenvalue equation:

The solution to this set of equations is, in general found by solving an nth
order

equation for l. Thus, in general, there can be up to n unique values for l,
with

an associated w vector for each of the n values.

In this case however, the matrix A has a special form since each row is a
constant

multiple of the first row. For such a matrix, the rank of the matrix is one,
and

all the eigenvalues of A are zero, except one. Since the sum of the eigenvalues

of a positive matrix is equal to the trace of the matrix (the sum of the
diagonal

elements), the non zero eigenvalue has a value of n, the size of the matrix.
This

eigenvalue is referred to as

.

Aside: The eigenvalue equation, when solved, produces one or more eigenvalues
with

associated eigenvectors. The word eigen, in German, means characteristic, or
one's

own. This is appropriate because eigenvalue equations arise naturally in many
instances

of mathematics and science -- on their own -- as a characteristic of the system
itself.

Eigenvalue equations are among the most frequent and universally occurring
concepts

in math and science; others include pi, the natural number e, Fibonacci numbers,

and the golden ratio. Even the Google search algorithm uses eigenvectors. (See
25

billion dollar eigenvector)