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Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for $k=0,\dots,l-1$. Define $M(\alpha,\beta)$ to be the $l\times l$ matrix whose entry in $i$th row and $j$th column is $\alpha_{i-1}^{j-1}\beta_{j-1}^{i-1}\zeta^{(i-1)(j-1)}$.

Question: Is $M(\alpha,\beta)$ nonsingular for all $\alpha,\beta$?

If $\alpha$ or $\beta$ is constant, then since $$ \alpha_i^j\beta_j^i\zeta^{ij}=\alpha_i^j(\beta_j\zeta^j)^i=\beta_j^i(\alpha_i\zeta^i)^j, $$ $M(\alpha,\beta)$ is equivalent to a Vondermonde matrix and thus nonsingular.

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  • $\begingroup$ My experiments suggest that this matrix can be singular---however, due to numerical concerns, am not yet conclusively claiming singularity. $\endgroup$
    – Suvrit
    Oct 8 '13 at 18:53
  • $\begingroup$ @suv....rit :Thanks! In your experiments, what is the smallest value of $l$ such that the matrix seems to be singular? $\endgroup$ Oct 9 '13 at 8:03
  • $\begingroup$ for me think broke at $l=40$, but should also break earlier---but I still suspect that this might be due to numerical roundoff (i.e., numerical rank was $l-1$) $\endgroup$
    – Suvrit
    Oct 9 '13 at 14:08
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    $\begingroup$ This question has the flavor of a theorem of Chebotarev on the non-singularity of the minors of the matrix with entry $\zeta^{ij}$ in position $(i,j)$, where $\zeta$ is a primitive $p$-th root of unity for a prime $p$. $\endgroup$ Oct 9 '13 at 14:23
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    $\begingroup$ Checking all possibilities for $\alpha$ and $\beta$, the answer is affirmative for all $\ell\le12$. $\endgroup$ Oct 10 '13 at 19:22

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