Extremal Rank Conditions and Their Linear Preservers
by
Olga Pshenitsyna
The Moscow State University

Let M_{m, n} be the set of m×n complex matrices, and let M_n=M_{m, n}. Suppose that M, N in M_n satisfy det(MN)=1. Then the mapping T :M_n --> M_n given by A --> MAN is linear and satisfies

det (T(A))= det (A)  for all A in Mn.

A linear operator T satisfying this equation is called a linear preserver of the determinant function or simply a determinant preserver. In 1897 it was proved by Frobenius that all bijective determinant preservers have the form A --> MAN  for all A in M_n(F) or A --> MA^tN  for all A in M_n(F), where A^t denotes the transposed matrix of A, and det(MN)=1. Linear preservers of different matrix invariants, properties, or relations have been the subject of intensive study from Frobenius's times till our days. During the last decades linear preservers of relations were the subject of most active investigation.

Namely, for any relation ~ on a set M_{m, n} of matrices over the field F we can define the linear preserver of ~ , i.e. those linear operators T satisfying

T (A) ~ T(B) whenever A ~ B.

For example, ~ could be commutativity, similarity, a certain order relation. Let us consider the following relation (*) on matrices:

rk (A|B)= rk (A)+ rk (B),

where (A|B) in M_{m, 2n} denotes the matrix formed by the set of columns of both matrices A, B in M_{m, n}. Among our results is the following

Theorem. If (*) be the relation described above, and a transformation T: M_{m, n}(F) --> M_{m, n}(F) be a linear bijective (*)-preserver, |F| > min(m, n), then transformation T is of standard form: T(X)=MXN  for all X in M_n(F), where M and N are invertible.

Date received: February 8, 2002