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?
> 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.
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):
"Jules" <julianro...@gmail.com> wrote in message news:1163398359.617744.163980@k70g2000cwa.googlegroups.com... > 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.
This is a discrete form of an integral equation. Viewing a(j), b(i,j), c(i) as step functions in i, j, you see you are solving
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?
For functional analysis to make sense of this sort of system you should specify Banach spaces X and Y for the sequences {a_j} and {c_i} respectively, such that (Tx)_i = sum_j b_{i,j} x_j gives a bounded linear map T from X into Y. You can then ask whether T is injective or surjective. However, checking these may not be easy in general. If T is a Fredholm operator of index 0, you have the Fredholm alternative: T x = y has a solution x for every y in Y if and only if the only solution to T x = 0 is x = 0.
Robert Israel isr...@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
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