I have a question about Siegel's Lemma (<https://en.wikipedia.org/wiki/Siegel%27s_lemma>) and bounding of underdetermined linear systems (n > m) of the form Ax = b. While the proof provides for an integer solution, is there a way to generalize for a real valued solution with the additional constraint that xi >= 0?



Bounded underdetermined linear system
With the little details you include in your question. I think you can use [Farkas Lemma](https://en.wikipedia.org/wiki/Farkas%27_lemma). It tell you that exactly one of the following two assertions is true: 1. There exists an ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$ such that ${\displaystyle \mathbf {Ax} =\mathbf {b} }$ and ${\displaystyle \mathbf {x} \geq 0}$. 2.There exists a ${\displaystyle \mathbf {y} \in \mathbb {R} ^{m}}$ such that ${\displaystyle \mathbf {A} ^{\mathsf {T}}\mathbf {y} \geq 0}$ and ${\displaystyle \mathbf {b} ^{\mathsf {T}}\mathbf {y} <0}$.


To bound each coordinates of $x$ you can use Gassian Elimination.


You can find more searching for "\(A^Ty\geq 0\)" on SearchOnMath.


**Edit:** If $m=n$. To bound each coordinates of $x$ as $$\left(a\sqrt{m}\right)^m,$$ you can apply apply [Hadamard's inequality](https://en.wikipedia.org/wiki/Hadamard%27s_inequality#Alternate_forms_and_corollaries) (please see this [thread](https://math.stackexchange.com/questions/2301590/hadamards-inequality-proof)) and [Cramer's rule](https://en.wikipedia.org/wiki/Cramer%27s_rule#General_case), where $$a=\max\_{ij}\{|a\_{ij}|,|b\_i|\}.$$


