Re: funny capacitor; almost-singular matrices

["John S. Denker" <jsd@MONMOUTH.COM>]

At 11:55 PM 3/4/01 -0500, I wrote:

> Inverting a singular matrix is like finding the intersection of two
> parallel lines.  If there is any inaccuracy in the description of the
> lines, they will intersect, but it will be someplace far away, and it won't
> mean anything.

Let me explain why I said "it won't mean anything".

Ludwik calculated the charges and voltages for his funny capacitor.  The
relationship is expressed by the vector/matrix equation
        Q = C V

with the particular values
-189             2.86   -0.46   -1.15   -1.25           -50
 143    =       -0.46    2.86   -1.15   -1.25   *        50
  65.2          -1.15   -1.15    3.26   -0.965           20
 -19.3          -1.25   -1.25   -0.965   3.46             0

This C matrix is almost singular.  If the calculation had been exact, the
matrix would have been exactly singular.

Ludwik emphasized his success in inverting this expression.  The result is
        B = inverse_of(C)
        V = B Q

with the particular values
-50              -99.641  -99.942 -100.005  -99.995     -189
 50     =        -99.942  -99.641 -100.005  -99.995  *   143
 20             -100.005 -100.005  -99.884 -100.116       65.2
6.82E-13         -99.995  -99.995 -100.116  -99.884      -19.3

The large matrix elements, and the small differences between them, are
scary.  But the result looks OK at first.  It looks OK until you realize
that the charge values in the rightmost column don't add up to zero.  If
you fix this, you get a very different picture.  For example, let's put the
leftover charge on object 4.  Using the same B matrix, and the same formula
        V = B Q

we get the particular values
-59.9995         -99.641  -99.942 -100.005  -99.995     -189
 40.00048 =      -99.942  -99.641 -100.005  -99.995  *   143
 9.988428       -100.005 -100.005  -99.884 -100.116       65.2
-9.98843         -99.995  -99.995 -100.116  -99.884      -19.2

Note that we are not successful at recovering the voltage pattern given the
charge pattern.  The slightest change (less than 1%) in the charge pattern
results in a catastrophic change in the voltage pattern.

Bottom line:  you have to be careful when dealing with singular and
almost-singular matrices.