Can the number of sign changes in a sequence of determinants tell us how many negative eigenvalues a symmetric matrix has? From notes, I've gathered that given a symmetric matrix, the number of sign changes in its characteristic polynomial is equal to the number of positive eigenvalues of $A$.

Proof: Let $p(x)$ be a real polynomial whose roots are all real. By Descarte’s rule,
the number $\sigma$ of positive eigenvalues is bounded by the number of sign changes in $p(x)$. Similarly, the number $\sigma'$ of negative eigenvalues is bounded by the number of sign changes in $p(−x)$. Hence the total number of positive and negative eigenvalues is bounded by $\sigma+\sigma'$ Now  $\sigma + \sigma' \leq n$ and the fact that all eigenvalues of a symmetric real matrix are real imply that the bound of Descarte’s rule of signs holds with equality. 

How can I use this proof to show that the sign changes in a certain sequence of determinants tells us how many negative eigenvalues $A$ has?
 A: I think that there are two true statements here: 
(1) If $A$ is an $n \times n$ real symmetric matrix, and $A_k$ denotes its $k \times k$ upper left corner, then the number of negative eigenvalues of $A$ is the number of sign changes in the sequence $(1, \det A_1, \det A_2, \ldots, \det A_n)$. 
(2) If $\det (A+z \mathrm{Id}) = a_0 + a_1 z + \cdots + a_{n-1} z^{n-1} + z^n$, then the number of negative eigenvalues of $A$ is the number of sign changes in the sequence $(a_0, a_1, \ldots, a_{n-1}, 1)$. 
The first statement is proved in your other question; the second is proved by Descartes as you say; I don't think there is any easy way to get from one to the other.

I've been thinking about this more, and there is a relation between the two questions. Let $\bigwedge\nolimits^k A$ be the $\binom{n}{k} \times \binom{n}{k}$ matrix describing the action of $A$ on $\bigwedge\nolimits^k \mathbb{R}^n$. Both the determinant of $A_k$ and the $k$-th coefficient of the characteristic polynomial are linear functions of $\bigwedge\nolimits^k A$. It would be neat to have some general theorem about sequences $f_1$, $f_2$, ..., $f_n$ where $f_k$ is a linear functional on $\mathrm{End}(\bigwedge\nolimits^k \mathbb{R}^n)$ so that the sign changes of $f_k(\bigwedge\nolimits^k A)$ compute signature.
