M-matrices: a generalization of M-matrices based on eventually nonnegative matrices.
Let ϕ be a surjective map on the space of n×n complex matrices such that r(ϕ(A)-ϕ(B))=r(A-B) for all A,B, where r(X) is the spectral radius of X. We show that ϕ must be a composition of five types of maps: translation, multiplication by a scalar of modulus one, complex conjugation, taking transpose and (simultaneous) similarity. In particular, ϕ is real linear up to a translation.
Consider —the ring of all upper triangular matrices defined over some field . A map is called a zero product preserver on in both directions if for all the condition is satisfied if and only if . In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map may act in any bijective way, whereas for the zero divisors and zero matrix one can write as a composition...