2
$\begingroup$

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?

  • 0
    @copper.hat - I said "in general" because q1 of the OP is general2012-09-19

1 Answers 1

6

The only real root of $x^5+x^3+x-3$ is $1$. Hence the only eigenvalue of $A$ is $1$, so $A$ is the identity matrix.

  • 0
    @Belgi: Being Hermitian ensures all of its eigenvalues are real.2012-09-19