On Cullis’ determinant for rectangular matrices

Abstract

We identify the column vectors of an n × k matrix ( k ⩽ n) with a k-tuple of vectors in the n dimensional vector space C n . The value of the alternative k-multiple linear functional D on the vector space of all n × k matrices is uniquely determined by the value on the finite subset { ( e i 1 , … , e i k ) ∣ i 1 < ⋯ < i k } of k-tuples of elements in the canonical basis { e 1, … , e n }. In [C.E. Cullis, Matrices and Determinoids, vol. 1, Cambridge University Press, 1913; vol. 2, 1918; vol. 3, 1925] Cullis called the value D( X) of the functional D at an n × k matrix X satisfying D ( e i 1 , … , e i k ) = ( - 1 ) ∑ ℓ = 1 k ( i ℓ - ℓ ) , the determinoid (which we call determinant throughout this paper) of X. In this article we study several properties of such matrices and give a characterization of the determinants by using the Laplace expansion property known for square matrices.