2
$\begingroup$

What are the necessary and sufficient condition(s) on a non-negative integral, invertible square $n \times n$ matrix $A$ to ensure that the unique solution of the matrix equation $A \mathbf{x} = \mathbf{1}$ lies in the interval $(0, \tfrac{1}{2}]^{n}$? Of course, Cramer's rule implies the condition $0 < 2 \det A_{j,1} \leq \det A$ for $1 \leq j \leq n$, where $A_{j,1}$ is the matrix $A$ with the $j^{\text{th}}$-column replaced with $\mathbf{1}$.

Farkas' Lemma gives conditions for positivity, but it is usually reserved for real matrices and seems a bit high powered for my question.

Thanks!

1 Answers 1

2

The inequality you derived from Cramer's rule is both necessary and sufficient. You can see this by verification (or disconfirmation) of the hypothesis on each component individually.

  • 1
    They would be logically and mathematically equivalent to your Cramer-based system of inequalities. It seems you might either be seeking a computationally more appealing form, or an alternate expression that is more usable for some other mathematical purposes. I'm uncertain as to how to look for or if there's a way to transform the conditions you have into a nontrivially distinct-looking alternative.2011-06-17