I find this exercise in my textbook.
Find all Hermitian matrices $A\in M_n(\mathbb{C})$ satisfying $A^5+A^3+A-3I=0$
I have two questions.
1) How do I solve a matrix polynomial? If I simply factorize it, I can only get those answers with the form $\lambda I$.
2) How a matrix being Hermitian (basically it means a matrix is "complexly" symmetric) makes it special in this problem?