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In article <1993Jan11.214251.12368@news.eng.convex.com>
dodson@convex.COM (Dave Dodson) writes:
>In article <1993Jan11.083253.9521@dutrun2.tudelft.nl>
>rcpshdb@dutrun2.tudelft.nl (Han de Bruijn) writes:
>>The condition number of a matrix  A  is defined as the absolute value of
>>its largest eigenvalue divided by its smallest eigenvalue. Right?
>>
>>I'm at loss how this is related to the loss of decimals when solving the
>>equations system:
>>                      A.x = b
>>
>>Can someone out there provide a simple argument?

  [Dave Dodson's good explaination deleted]

One thing that no one has remarked on: The original poster's reference
to *eigenvalues*.

    As Dave Dodson points out, the condition number of A is
norm(A) * norm(A-inverse).  No mention of eigenvalues.

    In fact, the condition number of a matrix has (almost) nothing to
do with the eigenvalues, as the example

        ( 1  t )                        -1   ( 1  -t )
    A = (      )   for t >> 1 shows:   A   = (       )
        ( 0  1 )                             ( 0   1 )

so the eigenvalues are both 1, but norm(A) = norm(A-inverse)
which are approximately  t, so the condition number is approx.
t**2.

    What the condition number *is* related to are the *singular values*.
For more info, look in a good book on numerical linear algebra
(e.g., Golub & van Loan's _Matrix_Computations_) on the "singular
value decomposition" (or SVD).  The SVD, once understood, makes
a lot of this a whole lot simpler to understand.


-- 
Alan McKenney        E-mail:  mckenney@cims.nyu.edu         (INTERNET)
Courant Institute,NYU,USA     ...!cmcl2!cims.nyu.edu!mckenney   (UUCP)