Matrix groups with positive spectra

AbstractThe main result of this paper is that every divisible matrix group with elements having positive spectra only is triangularizable. Such a group G is mapped into a Lie algebra L generated by real logarithms of the elements of G. It is shown that commutators of L are nilpotent and thus L is triangularizable by the Engel-Jacobson theorem. Since the logarithmic representation preserves invariant subspaces, the group G itself is triangularizable. In particular, every divisible matrix group whose elements are all similar to positive matrices is commutative and therefore similar to a group of positive matrices. A construction of the irreducible matrix groups with positive trace is given. Examples are presented to show that the divisibility condition cannot be dropped and that semigroups cannot replace groups even in the presence of divisibility