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Multiplicative maps on invertible matrices that preserve matricial properties. (English) Zbl 1042.15004

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.

MSC:

15A04 Linear transformations, semilinear transformations
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A18 Eigenvalues, singular values, and eigenvectors
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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