Re: capacitance: singular matrices

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

I wrote:
> > The capacitance matrix doesn't have an inverse.  It's singular.

...which is true.

Then at 11:43 PM 3/1/01 -0500, Ludwik Kowalski wrote:
> Aha, one more question.  What is wrong in saying that
>
>   V1=B11*Q1 + B12*Q2 + B13*Q3
>   V2=B21*Q1 + B22*Q2 + B23*Q3
>   V3=B31*Q1 + B32*Q2 + B33*Q3
>
> is the inverse of equation 2 (and vice versa)?

That begs the question of what values to use for the matrix elements Bij.

Folks may find it a bit of a challenge to find the inverse of any
capacitance matrix;  try directly inverting
.       3  -3   0
.      -3   5  -2
.       0  -2   2
if you dare.  Don't spend too much time on it.  You might want to "warm up"
by trying to invert
.       3  -3
.      -3   3

Again, be warned that this is the brute-force approach;  a more enlightened
approach is discussed below.

==========================================
Here's a better way to approach the problem:

1) Recall that for any given V, we know we can write a meaningful matrix
equation
        Q = C V                         (equation 7)
where C has been calculated (from Laplace's equation).  Therefore we can
find Q.

2) For given Q, it is questionable whether we can write
        V = B Q                         (equation 8)
for some matrix B = inverse_of(C).

3) So let's punt on question (2), and instead ask the question whether (for
known Q) there exists some V such that
        C V = Q                         (equation 9)

This allows us to ask an interesting question without having to exhibit the
inverse of C.

Since C is a singular matrix, this question (3) is profoundly analogous to
asking about whether (in Euclidean geometry) there is a point where two
parallel lines intersect.  The answer is problematic.  If we have two
distinct parallel lines, there is no such point.  If OTOH the two lines are
not distinct, i.e. they coincide, then we have an embarrassment of
riches:  the solution set contains not just "a point" but an infinite
number of points.

In our case, the analogy tells us that depending on what Q vector you
choose, there will either be NO voltage vector that produces that, or an
infinite number of different voltage vectors.

There is real physics behind this:
 -- If you choose a Q that doesn't satisfy charge conservation, there will
be no V that produces that.
 -- If there is one solution, gauge invariance guarantees there will be an
infinitude of solutions.

========================================
That's all for the physics.  Let's have some fun with the mathematics, to
see what goes wrong if you try to invert a singular matrix.

Again we will use the example matrix
.       3  -3   0
. C =  -3   5  -2                       (equation 10)
.       0  -2   2

You can try to invert that by hand, but you won't get very far.  You can
try to invert it using Excel, but that's not going to work, either.  So
let's ask a different question.  Let's take the inverse of
.       3+x  -3     0
. C'=  -3     5+x  -2                   (equation 11)
.       0    -2     2+x
(which is doable for any nonzero x) and then see what happens when x
becomes rather small.  This is done for you in the following spreadsheet:
  http://www.monmouth.com/~jsd/physics/singular-inverse.xls

The quantity "x" is called a regularizer.  If the regularizer is too big,
equation (11) is just too different from equation 10, and it doesn't tell
us anything about the physics problem we started with.  If the regularizer
is too small, the matrix elements of the inverse of (C') blow up, and that
isn't very interesting either.  Use the spreadsheet to play with various
values of x.

Amazingly enough, for intermediate (moderately small) values of x, the
inverse of C' is quite informative.

Let B' = inverse_of(C').  Consider
        V = B' Q                        (equation 12)

For any Q for which equation (9) has solutions, equation (12) will give you
(to a good approximation) an element of the solution set.  In fact, it will
give you the _smallest_ element, i.e. the one nearest the origin, i.e. the
one with the smallest |V|.  This may or may not be your favorite solution,
but it is a solution.

What's even more amazing is what happens when you feed equation (12) a
charge distribution that violates the laws of physics, i.e. cannot possibly
result from a solution of Laplace's equation because it doesn't satisfy
global charge neutrality.  For example, consider
.        .31
.  Q' = -.10
.       -.20

Then equation (12) will produce a voltage vector that corresponds to a
charge distribution that is "close" to Q'.

=========

You may be wondering how this regularization procedure can find "solutions"
to problems that don't have solutions.

The best way to think about it is to think of finding a value of V that
minimizes
        (Q - C V)^2
for given Q and C.  This minimization notion is helpful, because a minimum
exists, even when there is no Q for which Q=CV.

The regularizer transforms this into the related objective function
        (Q - C V - x V)^2
which includes a term in x^2 V^2.  This favors small |V|, other things
being equal.  This means there will be a unique minimum, rather than a
plethora of equally-good minima at wildly large |V|.