ELA, Volume 10, pp. 291-319, December 2003, abstract.

Multiplicative Maps on Invertible Matrices that 
Preserve Matricial Properties

Robert M. Guralnick, Chi-Kwong Li and Leiba Rodman

Descriptions are given of multiplicative maps on complex 
and real matrices that leave invariant a certain function,
property, or set of matrices: norms, spectrum, spectral 
radius, elementary symmetric functions of eigenvalues, 
certain functions of singular values, (p,q) numerical 
ranges and radii, sets of unitary, normal, or Hermitian 
matrices, as well as sets of Hermitian matrices with fixed 
inertia. The treatment of all these cases is unified, and 
is based on general group theoretic results concerning 
multiplicative maps of general and special linear groups, 
which in turn are based on classical results by Borel-Tits. 
Multiplicative maps that leave invariant elementary symmetric 
functions of eigenvalues and spectra are described also for 
matrices over a general commutative field.