Re: error: `eig' undefined near line 5 column 7

[Rich Shepard]

On Sat, 31 Jul 2004, Geraint Paul Bevan wrote:

> The eigenvalues (lambda) of a matrix (A) are the values of lambda that
> satisfy (lambda*x = A*x), where lambda is a scalar and I is the identity
> matrix. This can be rearranged to give det(lambda*I - A)=0, assuming
> that you are not interested in the trivial solution x=0.

Geraint,

  While it's been almost three decades since I took a course in linear
algebra, I do remember this much. :-)

> The answer that you are getting tells you the solutions to this equation
> and the fact that two of the roots are complex does not mean that there
> is anything wrong.

  True, but I am having difficulty interpreting the results. See below.

> As for what the answer is actually telling you, that really depends on
> what the original matrix describes. If the original problem described the
> equations of motion for a physical system, for example, the negative
> complex roots would tell you about a stable mode of oscillation which the
> system exhibits.

  The original matrix represents the results of pair-wise comparisons of
components in an environmental impact assessment. There is a fixed scale
(from 1 to 9) keyed to comparative labels such as "almost equal importance",
"slightly more important" and "greatly more important". The rows and columns
are the average values for the pair-wise comparisons. That's why the
diagonal is 1s and why the matrix is symetrical. In the example, if one pair
is scored as 3.000, then the mirror cell is 0.333.

  The way this is supposed to work as an Ordered Weighted Average (OWA) when
dealing with fuzzy logic, is that each element of the resultant eigenvector
is multiplied by the class of the matrix (i.e., the number of rows or
columns; 3 in the case of the example) to provide relative weights (in the
range [0,1]) for each component.

  This is why imaginary parts to the values don't make sense in the context
in which the computation is to be used. Working the example by hand -- using
an alternative to eigenvector calculation -- the numbers come out just fine.
Sigh.

Thanks,

Rich

-- 
Dr. Richard B. Shepard, President
Applied Ecosystem Services, Inc. (TM)
<http://www.appl-ecosys.com>



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