A double Sylvester determinant

Abstract

Given two $\left(n+1\right)\times \left(n+1\right)$-matrices $A$ and $B$ over a commutative ring, and some $k\in \left\{0,1,\dots ,n\right\}$, we consider the $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\times \left(\genfrac{}{}{0ex}{}{n}{k}\right)$-matrix $W$ whose entries are $\left(k+1\right)\times \left(k+1\right)$-minors of $A$ multiplied by corresponding $\left(k+1\right)\times \left(k+1\right)$-minors of $B$. Here we require the minors to use the last row and the last column (which is why we obtain an $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\times \left(\genfrac{}{}{0ex}{}{n}{k}\right)$-matrix, not an $\left(\genfrac{}{}{0ex}{}{n+1}{k+1}\right)\times \left(\genfrac{}{}{0ex}{}{n+1}{k+1}\right)$-matrix). We prove that the determinant $\det W$ is a multiple of $\det A$ if the $\left(n+1,n+1\right)$-th entry of $B$ is $0$. Furthermore, if the $\left(n+1,n+1\right)$-th entries of both $A$ and $B$ are $0$, then $\det W$ is a multiple of $\left(\det A\right)\left(\det B\right)$. This extends a previous result of Olver and the author ( arXiv:1802.02900 ).

arXiv comment16 pages. Slightly more detailed version available as ancillary file. Comments are welcome! v3 corrects some typos and updates references

References in corpus (1)

• The $n$ body matrix and its determinant

Cited by in corpus (1)

• The $n$ body matrix and its determinant