Linear maps preserving the Cullis' determinant. II

TL;DR

This paper characterizes linear maps that preserve the Cullis' determinant for certain rectangular matrices, extending previous work to cases where the preservation maps may be singular.

Contribution

It explicitly describes the structure of linear preservers of the Cullis' determinant for matrices with specific size and parity conditions, including singular maps.

Findings

Linear maps preserving the Cullis' determinant can be singular in certain cases.

Such maps are sums of two types: two-sided matrix multiplication and maps with equal-row matrices.

The results extend the understanding of linear preservers for matrices with odd sum of dimensions.

Abstract

This paper is the second 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. In this part we solve the linear preserver problem for the Cullis' determinant defined on the spaces of matrices of size $n\times k$ with $k \ge 4,\; n \ge k + 2$ and $n + k$ is odd. In comparison with the case when $n + k$ is even, in this case linear maps preserving the Cullis' determinant could be singular and are represented as a sum of two linear maps: first is two-sided matrix multiplication and second is any linear map whose image consists of matrices, all rows of which are equal.