Linear maps preserving the Cullis' determinant. I

TL;DR

This paper characterizes linear maps that preserve the Cullis' determinant for certain rectangular matrices, showing they are non-singular and can be expressed as two-sided matrix multiplications, extending classical determinant preservers.

Contribution

It provides the first explicit description of linear maps preserving the Cullis' determinant for matrices with specific size relations, expanding the linear preserver problem to new cases.

Findings

All such linear maps are non-singular.

They can be represented by two-sided matrix multiplication.

The case n=k or n=k+1 was previously solved.

Abstract

This paper is the first in the series of papers devoted to the explicit description of linear maps preserving the Cullis' determinant of rectangular matrices with entries belonging to an arbitrary ground field which is large enough. The Cullis' determinant is defined for every matrix of size $n\times k$, where $n \ge k \ge 1$ and is equal to the ordinary determinant if $n = k$. In this paper we solve the linear preserver problem for the Cullis' determinant for $k \ge 4, n \ge k + 2$ and $n + k$ is even. It appears that in this case all linear maps preserving the Cullis' determinant are non-singular and could be represented by two-sided matrix multiplication. Note that the cases where $n = k$ or $n = k + 1$ admit slightly different description allowing (sub)matrix transposition and were completely studied before: the case where $n = k$ is a classical linear preserver problem for the ordinary determinant and was solved by Frobenius; the complete characterisation for the case where $n = k + 1$ was obtained in the previous paper by the authors.