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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?

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    There is no general way by abel-ruffini (for the 'easy' case $n=1$). In this case try Jordan normal form, maybe it will help2012-09-19
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    Hermitian matrices are diagonalizable, so this reduces to a question about scalars satisfying the given polynomial constraint.2012-09-19
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    @Belgi: That is not true, $x=1$ satisfies the equation, reducing to $(x-1)(x^4+x^3+2x^2+2x+3)$ wihch can be solved :-).2012-09-19
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    @copper.hat - I said "in general" because q1 of the OP is general2012-09-19

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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.

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    but the matrix can have non-real eigenvalues to, no ?2012-09-19
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    @Belgi: Being Hermitian ensures all of its eigenvalues are real.2012-09-19