Message from discussion Infinite system of linear equations

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Date: Mon, 13 Nov 2006 10:09:48 +0100
From: Han de Bruijn <Han.deBru...@DTO.TUDelft.NL>
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Subject: Re: Infinite system of linear equations
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Jules wrote:

> Suppose we have a doubly-indexed sequence {b_(i, j)} of real numbers,
> with i, j positive integers.  Suppose also that we have a sequence
> {c_i} of reals.  We wish to find a sequence {a_j} of reals so that the
> sum as j goes from 1 to infinity of b_(i, j) * a_j = c_i for each i.
> This is, in some sense, a collection of countably-many linear equations
> in countably-many variables.  Are there any conditions on the
> coefficients b_(i, j) that would guarantee existence and/or uniqueness
> of a solution {a_j}?  If there were only finitely-many equations and
> variables (the same number of each), then one could simply check that
> the determinant of the coefficient matrix is non-zero.  Is there any
> analog of determinant for an "omega-by-omega" matrix?

Special cases of such infinite systems of linear equations have been
investigated. In "Multigrid Calculus" it is proved that the infinite
tri-diagonal system of linear equations:
   ........
      -b  1  -a
          -b   1  -a
              -b   1  -a
                   .........
is equivalent with a second order ordinary differential equation:

     d^2u/dx^2 - P.du/dx + (P^2-Q^2)/4.u = 0

where the matrix coefficients - with infinitesimal grid-spacings (dx) -
are given by:

     a = exp(-(P-Q)/2.dx)  ;  b = exp(+(P+Q)/2.dx)

And the general solution is of the form (with A,B arbitrary):

     u(x) = A.exp((P-Q)/2.x) + B.exp((P+Q)/2.x)

The key reference is:

     http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/calculus.pdf

Han de Bruijn