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Consider a complex matrix $A\in\mathbb{C}^{n+1\times m}$ such that $$ A=\begin{bmatrix} 1 & 1 & \dots & 1\\ Bc_1 & Bc_2 & \dots & Bc_{m} \end{bmatrix}, $$ where $c_1,\dots ,c_m\in\mathbb{C}^p$ are fixed and $B\in\mathbb{C}^{p\times n}$ is such that $$ (Bc_k)_j = \sum_{\ell=1}^{p}(c_k)_\ell e^{i\lambda_{k,\ell} j}, $$ with $\lambda_{k,\ell}\in\mathbb{C}$ for all $k\in \{1,\dots, m\}$ and $\ell\in\{1,\dots ,p\}$. (In the latter expression $j$ is an index and $i$ is the imaginary unit).

I am looking for a closed form of the kernel of the matrix $A$. I think that working on the determinant might be useful.

As a first step, I am working on the special case where $p=1$. In this case, ommiting the $\ell$ subscript from $\lambda$, we have $$ \begin{bmatrix} 1 & 1 & \dots & 1\\ c_1e^{i\lambda_1} & c_2e^{i\lambda_2} & \dots & c_me^{i\lambda_m}\\ c_1e^{i2\lambda_1} & c_2e^{i2\lambda_2} & \dots & c_me^{i2\lambda_m}\\ \vdots & \vdots & &\vdots\\ c_1e^{in\lambda_1} & c_2e^{in\lambda_2} & \dots & c_me^{in\lambda_m}\\ \end{bmatrix}, $$ which is very similar to a Vandermonde matrix. Note that in this case we can, for each column $k\in\{1,\dots ,m\}$, multiply by one by applying a column-multiplication elementary matrix which multiplies by $z_k:=c_ke^{i\lambda_k}$ and an other which multiplies by $z_k^{-1}$. Therefore, by the Laplace expansion over the first row and the Vandermonde structure of the related submatrices one obtains $$ det(A)=\left[\prod_{k=1}^mc_ke^{i\lambda_k}\right]\cdot\left[\sum_{k=1}^m\frac{(-1)^{k+1}}{c_ke^{i\lambda_k}}\prod_{(s_1,s_2)\in S_k}(e^{i\lambda_{s_1}}- e^{i\lambda_{s_2}})\right], $$ where $S_k=\{(s_1,s_2)\in [m]^2:s_1<s_2, s_1\neq k, s_2\neq k\}$. Even though this is a closed form expression, it is not precisely informative for the kernel of $A$.

Any comments or suggestions will be greatly appreciated.

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  • $\begingroup$ A similar problem is discussed in mathoverflow.net/questions/236323, see also arxiv.org/abs/2103.10776 (J. Phys. A: Math. Theor. 54, 375201 (2021)). $\endgroup$
    – Fred Hucht
    2 days ago
  • $\begingroup$ Thank you, Fred, for your comment. I have read your post and the Appendix B of your paper, it is very insightful. I see that you're able to reduce the problem of computing the determinant of the Vandermonde type matrix into the problem of computing the determinant of a Hankel matrix. However, unfortunately I don't see how to obtain a benefit by doing something similar in my case. How would you use that to have an explicit form for the kernel of the matrix A? $\endgroup$
    – PIII
    yesterday

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