Infinite system of linear equations - sci.math

Message from discussion Infinite system of linear equations
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From: Gottfried Helms <he...@uni-kassel.de>
Newsgroups: sci.math
Subject: Re: Infinite system of linear equations
Date: Mon, 13 Nov 2006 07:30:44 +0100
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Am 13.11.2006 07:12 schrieb Jules:
> 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?
> 

Just an idea:

I actually don't know; what comes to mind is:

 - either the matrix is triangularizable by noniterative
   similarity transform
 - or you can find eigenvectors - and the associated
   (matrix of) Eigenvalues.

Then the determinant is the limit of the product of the
diagonal-entries.

But I think this implies strict conditions for your
infinite matrix.

Gottfried Helms