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A double Sylvester determinant
arXiv.org, v 20(2), pp 261-274
25 Aug 2020
Abstract
Given two \(\left( n+1\right) \times\left( n+1\right)\)-matrices \(A\) and \(B\) over a commutative ring, and some \(k\in\left\{ 0,1,\ldots,n\right\}\), we consider the \(\dbinom{n}{k}\times\dbinom{n}{k}\)-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 \(\dbinom{n}{k}\times\dbinom{n}{k}\)-matrix, not an \(\dbinom{n+1}{k+1}\times\dbinom{n+1}{k+1}\)-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 ).
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Details
- Title
- A double Sylvester determinant
- Creators
- Darij Grinberg - Drexel University
- Publication Details
- arXiv.org, v 20(2), pp 261-274
- Publisher
- Cornell University Library, arXiv.org; Ithaca
- Resource Type
- Other
- Language
- English
- Academic Unit
- Mathematics
- Scopus ID
- 2-s2.0-85126311407
- Other Identifier
- 991019173980104721