🚀 go-pugleaf

mckenney@cims.nyu.edu (Alan McKenney) writes:


>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*.

The singular values of A are the square roots of the eigenvalues of
the symmetric matrix AA'.  These are of course the eigenvalues of A
itself whenever A is symmetric and positive semidefinite.  So saying
that the condition number has (almost) nothing to do with the
eigenvalues is not accurate either.  The original poster may have had
the symmetric positive semidefinite situation in mind.

The eigenvalues of AA' also have an exact relationship to the norm of
A if we define the latter appropriately.  For example, if we define
the norm of A as:
                   |Ax|
	|A| = max  ---- 
               x    |x|

Then we get 

|A| = square root of the largest eigenvalue of AA' and

|Inverse(A)| = 1 / (square root of the smallest eigenvalue of AA')

and the condition number is the product of these two norms.
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209