3 M needs to be positive, not just hermitian

Assume $M$ is an invertible Hermitian positive matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues ${ \lambda_n }$is

$H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$.

In principle, the eigenvalues are encoded in the characteristic polynomial

$\phi_t (M) = \mathrm{det}(tI-M) = \prod_n(t-\lambda_n) = t^{N} + a_{N-1} t^{N-1} \cdots + a_1 t +a_0$.

The trace $\mathrm{Tr}\, M$ is given by the coefficient $a_{N-1}$ in the characteristic polynomial:

$\lim_{t \to \infty} \frac{\phi_t (M)-t^N}{t^{N-1}} = a_{N-1} = (-1)^{N-1} \, \mathrm{Tr}\, M = (-1)^{N-1} \sum_n \lambda_n$.

Is there a similar relationship between the entropy and the characteristic polynomial?

Assume $M$ is an invertible Hermitian matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues ${ \lambda_n }$is

$H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$.

In principle, the eigenvalues are encoded in the characteristic polynomial

$\phi_t (M) = \mathrm{det}(tI-M) = \prod_n(t-\lambda_n) = t^{N} + a_{N-1} t^{N-1} \cdots + a_1 t +a_0$.

The trace $\mathrm{Tr}\, M$ is given by the coefficient $a_{N-1}$ in the characteristic polynomial:

$\lim_{t \to \infty} \frac{\phi_t (M)-t^N}{t^{N-1}} = a_{N-1} = (-1)^{N-1} \, \mathrm{Tr}\, M$.= (-1)^{N-1} \sum_n \lambda_n$. Is there a similar relationship between the entropy and the characteristic polynomial? 1 # Is there a simple relation between the entropy of a matrix and its characteristic polynomial? The Von Neumann entropy$H$of a matrix$M$with eigenvalues${ \lambda_n }$is$H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$. In principle, the eigenvalues are encoded in the characteristic polynomial$\phi_t (M) = \mathrm{det}(tI-M) = \prod_n(t-\lambda_n) = t^{N} + a_{N-1} t^{N-1} \cdots + a_1 t +a_0 $. The trace$\mathrm{Tr}\, M$is given by the coefficient$a_{N-1}$in the characteristic polynomial:$\lim_{t \to \infty} \frac{\phi_t (M)-t^N}{t^{N-1}} = a_{N-1} = (-1)^{N-1} \, \mathrm{Tr}\, M \$.

Is there a similar relationship between the entropy and the characteristic polynomial?