Let $A$ be a hermitian matrix with the characteristic polynomial $p_A=a_0+a_1x+...+a_nx^n.$ Furthermore let $p$ be the number of sign changes in the sequence $\{a_0,a_1,...a_{n-1},1\}$ and $q$ be the number of sign changes in the sequence $\{a_0,-a_1,...(-1)^{n-1}a_{n-1},(-1)^n\}$. Show that the signature of $A$ is given by $(p,q)$.
I don't know if this is a good way to go about it but upon seeing it my first thought was to make use of Rolle's theorem from Analysis. Since the signature of a hermitian matrix is equal to the number of positive and negative eigenvalues observing the zeros of the characteristic polynomial would be the way to go. By Rolle's theorem if $\lambda_1<\lambda_2$ are two zeros of the polynomial then there exists $\mu$ with the properties $p_A'(\mu)=0$ and $\lambda_1<\mu<\lambda_2$. I also assumed that an $n$th root of $p_A$ would also be a $n-1$th zero of $p_A'$. I think this is good starter for a proof but I'm not sure how to go on from here.
