The Cullis' determinant as Pfaffian

TL;DR

This paper introduces a novel expression of the Cullis' determinant as a Pfaffian and presents an efficient polynomial-time algorithm for its computation.

Contribution

It expresses the Cullis' determinant as a Pfaffian and develops a new polynomial-time algorithm for calculating it.

Findings

Cullis' determinant expressed as a Pfaffian.

Efficient polynomial-time algorithm for calculation.

Generalizes determinant concept for rectangular matrices.

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 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 algorithm for calculating the Cullis' determinant of given matrix.