# Thread:

1. ## 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.

2. ## Re: solution to infinite dimensional matrix

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:

$A = \begin{bmatrix}1&-1&-1&-1&\ldots\\ 0&1&-1&-1&\ldots\\ 0&0&1&-1&\ldots\\ \vdots&\vdots&&\ddots&\ddots\end{bmatrix}$,

with 1s on the main diagonal, –1s everywhere above it and 0s everywhere below it. Then Ax=0, where $x = \bigl(\tfrac12,\tfrac14,\tfrac18,\ldots,2^{-n},\ldots\bigr).$
But if Ty = 0 with $y_n=0$ whenever n≥N, then by looking at the (N–1)th coordinate of Ay you can see that $y_{N-1}=0$, and by "backwards induction" $y_n=0$ for all n.

3. ## Re: solution to infinite dimensional matrix

THANK YOU SO MUCH!!!

That helps a lot.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9285057187080383, "perplexity_flag": "middle"}