<s>[INST] Hermitian matrix such that $4M^5+2M^3+M=7I_n$ $n$ is a positive integer. Besides the identity matrix $I_n$, does there exist other  $n\times n$   Hermitian matrix $M$, such that the following equality
$$4M^5+2M^3+M=7I_n $$
hold?
I try this: Since  $4M^5+2M^3+M=7I_n$, then 
$$(M-I_n)(4M^4+4M^3+6M^2+6M+7I_n)=0$$
but, What should I do next? Thanks!
 [/INST] Hermitian matrices are diagonalizable and they have only real eigenvalues. Let
$$
M=H^{-1}\mathrm{diag}(d_1,\ldots,d_n)H,
$$
where $D=\mathrm{diag}(d_1,\ldots,d_n)$ is the diagonal matrix which contains the eigenvalues of $M$. Note that
$$
4x^4+4x^3+6x^2+6x+7= (2x^2+1)^2+3(x+1)^2+2x^2+4 \ge 5,
$$
for all real $x$. This implies that
$$
L=4M^4+4M^3+6M^2+6M+7= H^{-1}\mathrm{diag}(r_1,\ldots,r_n)H,
$$
with $r_i\ge 5$, for all $i$. Thus $L$ is invertible. Hence
$$
(4M^4+4M^3+6M^2+6M+7)(M-I)=0,
$$
implies that $M=I$!
</s>