There are a bunch of matrix properties that ensure that all the eigenvalues are real, chief among them being symmetric. However, I cannot find nontrivial examples of semigroups (under any kind of product) of matrices all with real eigenvalues. Examples include, I believe:
- Semigroup under multiplication generated by some matrices (with real eigenvalues) that commute.
- Semigroup under multiplication generated by a symmetric matrix and a symmetric positive definite matrix.
Question: Is there a nontrivial semigroup of real matrices, all with real eigenvalues, that does not fall in these two examples?
What about upper triangular matrices (with real diagonal entries)?