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
)



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