Analytic hierarchy process (AHP), what is the significance of eigenvalues/eigenvectors? I was to a new years party today. Mathematics interested as I am I tried to discuss
eigenvalues with my friend there. He did not know what eigenvalues are but said
that he has heard about them in his research in something called
Analytic Hierarchy Process (AHP).
Analytic hierarchy process, wikipedia link
I found the following passage in wikipedia that mentions the word eigenvector, and the reference to Oskar Perron:
"Non-Monotony of some weight extraction methods
Within a comparison matrix one may replace a judgement with a less favourable judgement and then check to see if the indication of the new priority becomes less favourable then the original priority. In the context of tournament matrices, it has been proven by Oskar Perron in,[32] that the principle right eigenvector method is not monotonic. This behaviour can also be demonstrated for reciprocal n x n matrices, where n > 3. Alternative approaches are discussed in.[33][34][35]"
Reading about Oskar Perron I found that he worked in differential equations. Then reading about differential equations and eigenvalues in my book Advanced Engineering Mathematics, in the chapter Eigenvectors from the Start, page 299, I found:
"The general solution of system of differential equations $x' = Ax$ is known as soon as the eigenvalues and eigenvectors are known. In fact, if the eigenvalues are $\lambda_1$, $\lambda_2$
and the corresponding eigenvectors are $v_1, v_2$ then the general solution would be
$$x = a \cdot \exp(\lambda_1 \cdot t) \cdot v_1 + b \cdot \exp(\lambda_2 \cdot t) \cdot v_2$$
So my question is: What is the significance of eigenvalues in Analytic Hierarchy Process (AHP)?

The second link by Amzoti in the comment above is very clear about how the algorithm goes.
Illustration of Analytic hierarchy process
I have begun to study this by looking at the more arithmetic table $n/k$ as the pairwise comparison:
$$\begin{array}{llllll}
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\
 2 & 1 & \frac{2}{3} & \frac{1}{2} & \frac{2}{5} & \frac{1}{3} \\
 3 & \frac{3}{2} & 1 & \frac{3}{4} & \frac{3}{5} & \frac{1}{2} \\
 4 & 2 & \frac{4}{3} & 1 & \frac{4}{5} & \frac{2}{3} \\
 5 & \frac{5}{2} & \frac{5}{3} & \frac{5}{4} & 1 & \frac{5}{6} \\
 6 & 3 & 2 & \frac{3}{2} & \frac{6}{5} & 1
\end{array}$$
Mathematica:
TableForm[Table[
  Rationalize[
   Total[Transpose[
      MatrixPower[N[Table[Table[n/k, {k, 1, i}], {n, 1, i}]], i]]]/
    Total[Total[
      Transpose[
       MatrixPower[N[Table[Table[n/k, {k, 1, i}], {n, 1, i}]], 
        i]]]]], {i, 1, 6}]]

The output is very orderly:
$$\begin{array}{llllll}
 1 & \text{} & \text{} & \text{} & \text{} & \text{} \\
 \frac{1}{3} & \frac{2}{3} & \text{} & \text{} & \text{} & \text{} \\
 \frac{1}{6} & \frac{1}{3} & \frac{1}{2} & \text{} & \text{} & \text{} \\
 \frac{1}{10} & \frac{1}{5} & \frac{3}{10} & \frac{2}{5} & \text{} & \text{} \\
 \frac{1}{15} & \frac{2}{15} & \frac{1}{5} & \frac{4}{15} & \frac{1}{3} & \text{} \\
 \frac{1}{21} & \frac{2}{21} & \frac{1}{7} & \frac{4}{21} & \frac{5}{21} & \frac{2}{7}
\end{array}$$
which is $$\frac{k}{n(n+1)/2}$$
I will continue editing this later when I have the time.
nn = 5
Do[
 A = N[Table[Table[n/k, {k, 1, nn}], {n, 1, nn}]];
 A = MatrixPower[A, n];
 MatrixForm[N[A]];
 MatrixForm[Total[Transpose[N[A]]]];
 Total[Total[Transpose[N[A]]]];
 Print[Rationalize[
   Total[Transpose[N[A]]]/Total[Total[Transpose[N[A]]]]]], {n, 1, 6}]

