Let $n$ be any given positive integer. For any nonempty disjoint subset $A,B\subseteq \{1,2,\cdots ,n\}$,does there exist some specific matrix $M$ which is similar to the Vandermonde matrix such that $$\begin{vmatrix} M \end{vmatrix}=\prod\limits_{i\in A,j\in B}(x_j-x_i)$$ $$\text{or}$$$$\begin{vmatrix} M \end{vmatrix}\neq 0 \quad\text{if and only if}\prod\limits_{i\in A,j\in B}(x_j-x_i)\neq 0\quad?$$
$\begingroup$
$\endgroup$
3
-
5$\begingroup$ What surely works is the Sylvester matrix of the monic polynomial $\prod_{i\in A}\left(X - x_i\right)$ and the monic polynomial $\prod_{j\in B}\left(X - x_j\right)$". Its entries will be certain (minus-plus) elementary symmetric functions of the x'es. $\endgroup$– darij grinbergJul 12, 2015 at 15:27
-
$\begingroup$ Also I think there is a dissonance in the question: Either $A$ and $B$ should be subsets of $\left\{1,2,\ldots,n\right\}$ (rather than of the set of pairs you wrote), or the products should run over $\left(i,j\right)\in A$. I supposed you want the former. $\endgroup$– darij grinbergJul 12, 2015 at 15:28
-
$\begingroup$ Yes,yes,darij grinberg,thank you very much! Last night I was in such a hurry that I miswrote the question! I have modified the question! $\endgroup$– user173856Jul 13, 2015 at 5:27
Add a comment
|