Thread: solution to infinite dimensional matrix

Consider the equations Ax=0 where both the number of columns and rows of A are countably infinite and all entries are either 1, 0 or -1.

Is the following statement true or false?

Ax=0 has a nonnegative bounded solution (i.e., Ax=0 for some x=(x1,x2,...) with xi>=0 for all i and sum_i(xi)<infinity)

iff

Ax=0 has a nonnegative bounded solution with at most finitely many nonzeros (i.e., Ax=0 for some x=(x1,x2,...) with xi>=0 for all i and sum_i(xi)<infinity AND xi=0 for all but finitely many i).

I guess the answer is no.

I greatly appreciate any reference. I have checked a few books but can't find the answer.

Originally Posted by vivian6606

Consider the equations Ax=0 where both the number of columns and rows of A are countably infinite and all entries are either 1, 0 or -1.

Is the following statement true or false?

Ax=0 has a nonnegative bounded solution (i.e., Ax=0 for some x=(x1,x2,...) with xi>=0 for all i and sum_i(xi)<infinity)

iff

Ax=0 has a nonnegative bounded solution with at most finitely many nonzeros (i.e., Ax=0 for some x=(x1,x2,...) with xi>=0 for all i and sum_i(xi)<infinity AND xi=0 for all but finitely many i).

I guess the answer is no.

I greatly appreciate any reference. I have checked a few books but can't find the answer.

Suppose that A looks like this:

,

with 1s on the main diagonal, –1s everywhere above it and 0s everywhere below it. Then Ax=0, where
But if Ty = 0 with whenever n≥N, then by looking at the (N–1)th coordinate of Ay you can see that , and by "backwards induction" for all n.

THANK YOU SO MUCH!!!

That helps a lot.