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.