Paper
Linear maps preserving the Cullis' determinant. I
Authors
Alexander Guterman, Andrey Yurkov
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 , where and is equal to the ordinary determinant if . In this paper we solve the linear preserver problem for the Cullis' determinant for and 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 or admit slightly different description allowing (sub)matrix transposition and were completely studied before: the case where is a classical linear preserver problem for the ordinary determinant and was solved by Frobenius; the complete characterisation for the case where was obtained in the previous paper by the authors.