Linear maps preserving the Cullis’ determinant

The notion of determinant of square matrix has been studied in many contexts and one of them is the investigation of linear maps preserving the determinant. The first result in this direction dates back to 1897 and is due to Frobenius. He established in [2] that every such linear map should be a composition of two-sided matrix multiplication X →AXB with det(AB) = 1 and matrix transposition X →X^t.

This result prompted the investigation of so-called linear preserver problems concerning the characterization of linear operators on matrix spaces that leave certain functions, subsets, relations, etc., invariant.

The research started by Frobenius was continued in the works by Schur, Dieudonn´e, Dynkin and has since grown into substantial and rapidly advancing field within mathematical science.

The Cullis’ determinant is a generalization of the notion of the classical determinant to the set of rectangular n×k-matrices with n ≥k which is introduced in [1]. It could be expressed as an alternating sum of minors of a matrix and has several properties similar to the classical determinant. For instance, the Cullis’ determinant is multilinear with respect to matrix columns and satisfies the Laplace expansion theorem along every column as it is shown in [3].

A question about generalization of the aforementioned Frobenius’ theorem to the Cullis’ determinant arises naturally. We obtain the answer to this question and establish a complete characterization of linear maps preserving Cullis’ determinant. It appears that the answer depends on k and the parity of n + k. In particular, if n ≥k +2 and n+k is even, then every linear map preserving the Cullis’ determinant is non-singular and has the form X →AXB with square matrices A, B of size n ×n and k ×k satisfying certain additional condition. The answer for all the rest cases of n and k will be presented as well. If time permits, we will also discuss the group structure and possible generating sets of the corresponding group.

The talk is based on the joint work with Alexander Guterman.

References

[1] C. E. Cullis. Matrices and Determinoids: Volume 1. Calcutta University Readership Lectures. Cambridge University Press, 1913.

[2] F. G. Frobenius. Uber die darstellung der endlichen gruppen durch lineare substitutionen. Sitzungsberichte der Deutsche Akademie der Wissenschaften zu Berlin: 944–1015, 1897.

[3] Y. Nakagami and H. Yanai. On Cullis’ determinant for rectangular matrices. Linear Algebra and its Applications, 422(2):422–441, 2007.