On Cullis' determinant for rectangular matrices
In 1913-1925, C. E. Cullis published a three volume treatise called Matrices and Determinoids, in which he defined what here is called a determinant for \(n\times k\) matrices, \(k\leq n\) and studied in the context of Grassmann algebras. In the authors' notation let \(S_k^I\) be the set of injections from \(K=\{1,\dots,k\}\) into \(I=\{1,\dots,n\},\) and \({\mathcal C}_k^I\) be the family of \(k\)-element subsets of \(I\). For an increasingly reordered set \(c=\{i_1<\dots<i_k\}\in {\mathcal C}_k^I\) define its sign as \(\text{sgn}_I(c)=(-1)^{\sum_{l=1}^k (i_l-l)}.\) For \(\sigma \in S_k^I\) define \(\text{sgn}_{nk} (\sigma)=\) sign of the set \(\sigma(K)\) \(\times\) sign of permutation \((\sigma(1),\dots,\sigma(k))\) relative to the increasing rearrangement of \(\sigma(K).\) Let \(\{e_1,\dots,e_n\}\) be the standard basis of \(\mathbb C^n,\) and let \( \bigwedge \mathbb C^n =\bigoplus_{k=0}^n \bigwedge^k \mathbb C^n\) be the Grassmann algebra based on it. On \(\bigwedge^k \mathbb C^n\) we have the usual inner product \((x_1\wedge \cdots \wedge x_k| y_1\wedge \cdots \wedge y_k)=\det ((x_i| y_j)),\) and given an \(n\times k\) matrix \(B,\) we have the linear operator \(\Gamma(B)\) defined by \(\Gamma(B)(x_1\wedge \cdots \wedge x_l)= Bx_1\wedge \cdots \wedge Bx_l,\) \(l=1,\dots,k.\) Given \(c \in {\mathcal C}_k^I,\) choose a \(\sigma \in S_k^I, \) with \(\sigma(K)=c\) and define \(e_c^I= \text{sgn}_{nk} (\sigma) e_{\sigma(1)}\wedge \cdots \wedge e_{\sigma(k)};\) a \(k\)-vector independent of \(\sigma.\) For \(1\leq k\leq n\) and \(K=\{1,2,\ldots, k\},\) and an \(n\times k\) matrix \(X\), the Cullis- and co-Cullis determinants are defined respectively by \[ \det_{nk}(X)=\sum_c (\Gamma(X)e_K^K | e_c^I), \quad \text{Det}_{nk}(X)=\sum_c (\Gamma(X)e_K^K | \text{sgn}_I(c) e_c^I). \] Both these determinants revert to the usual one if \(k=n\) and are alternative \(k\)-linear forms. In later sections the Cullis determinant is characterized, several expansions are given, and formulae reminiscent of the the Cauchy-Binet expansion are proved for \(\det_{nl}(XY).\) The reviewer remarks that there exist other generalizations of the determinant to (real) rectangular matrices: for example the notion of matrix volume.
scientific article; zbMATH DE number 5141623