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Proving or disproving this matrix V is invertible.

kaienfr

New member
Jul 16, 2020
3
Let us take $n\in \mathbb{N}^*$ bins and $d\in \mathbb{N}^*$ balls. Denote the set $B = \{\alpha^1, \ldots, \alpha^m\}$ to be all possible choices for putting $d$ balls into $n$ bins, such as
$$\alpha^1 = (d,0,\ldots, 0), ~ \alpha^2 = (0,d,\ldots, 0), \ldots$$
Let us define the matrix $V$ as:
$$V = \begin{bmatrix}
(\alpha^1)^{\alpha^1} & \cdots & (\alpha^1)^{\alpha^m}\\
(\alpha^2)^{\alpha^1} & \cdots & (\alpha^2)^{\alpha^m}\\
\vdots & \vdots & \vdots\\
(\alpha^m)^{\alpha^1} & \cdots & (\alpha^m)^{\alpha^m}
\end{bmatrix}$$
where the notation $(\alpha^i)^{\alpha^j} = \displaystyle\prod_{k=1}^{n}(\alpha^i_k)^{\alpha^j_k}$, under assumption that $0^0=1$.

Now, I'd like to know: "is the matrix $V$ invertible?"

I have tested several examples, and it seems that $V$ is always invertible, but I have no idea how to prove it or find a counterexample.
Here are two facts I understood:
  1. all diagonal elements are strictly positives, so the trace of $V$ is strictly positive.
  2. the matrix $V$ is not symmetric.
Does anyone can help to prove or disprove the invertibility of $V$? Thanks a lot in advance for sharing any idea.