Abstract:The Cullis' determinant is a generalization of the ordinary determinant for rectangular matrices. It is defined as the alternating sum of maximal minors of a given matrix. In this paper we express the Cullis' determinant of a matrix $X$ as the Pfaffian of the matrix obtained from $X$ by matrix multiplication and transposition.
Relying on this result, we present an efficient polynomial-time division-free algorithm for calculating the Cullis' determinant of a given matrix with entries belonging to the commutative ring. We provide an asymptotical analysis of its arithmetical complexity in comparison to the definition-based algorithm.
In addition, we derive formulas for horizontal expansion of the Cullis' determinant which complements the existing formula for Laplace expansion along the columns of a matrix.
From: Andrey Yurkov [view email]
[v1]
Wed, 13 May 2026 18:21:43 UTC (15 KB)
[v2]
Thu, 2 Jul 2026 09:29:53 UTC (49 KB)
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