Sign up ×  Let $A\overline{X}=0$, where $A=(a_{i,j})\in M_{L,
n}(\mathbb{C})$ is a given non-zero $L\times n$ matrix over the
complex numbers $\mathbb{C}$, and $\overline{X}=(x_1, x_2,
\cdots, x_n)^T $ is an $n$-column vector.  Suppose $L< n$, then
we know that the above equations have non-zero solutions. My
question is:  What are the sufficient (and necessary) conditions
on $A$ such that we can find one solution $\overline{X}$
satisfying:  $$(P)\ \ \ \ \ \ \ \ |x_1|>\sum_{i=2}^n|x_i|$$
Here are some observations and remarks:    1. We may assume
$L>1$, since for $L=1$, $|a_{1,1}|<|a_{1,i}|$ for some $i>1$ is
a sufficient condition.    2. For $a_{i,1}\neq 0$, since
$x_1=-\sum_{j=2}^n\frac{a_{i,j}x_j}{a_{i,1}}$, we know that a
necessary condition for the property $(P)$ to be hold is that
$\exists j>1$, s.t., $|a_{i,j}|>|a_{i,1}|$.    3. My primary
motivation to ask this question is that I want to consider
equations in the general group algebras $\mathbb{Z}G$ for some
countable discrete group $G$; see this prob