Determinants of block matrices with noncommuting blocks

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• 作者：Nat Sothanaphan ^{natsothanaphan@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A15
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：512
• 期：Complete
• 页码：202-218
• 全文大小：336 K

文摘

Let M   be an mn×mn$mn\times mn$ matrix over a commutative ring R. Divide M   into bbdeef64b93c" title="Click to view the MathML source">m×m$m\times m$ blocks. Assume that the blocks commute pairwise. Consider the following two procedures: (1) Evaluate the n×n$n\times n$ determinant formula at these blocks to obtain an bbdeef64b93c" title="Click to view the MathML source">m×m$m\times m$ matrix, and take the determinant again to obtain an element of R  ; (2) Take the mn×mn$mn\times mn$ determinant of M. It is known that the two procedures give the same element of R. We prove that if only certain pairs of blocks of M commute, then the two procedures still give the same element of R, for a suitable definition of noncommutative determinants. We also derive from our result further collections of commutativity conditions that imply this equality of determinants, and we prove that our original condition is optimal under a particular constraint.