The question I am working on is as follows:
Let $A$ be an $n$ by $n$ Hermitian matrix and let, for $j \in [1, n]$, $A_j$ be the submatrix consisting of the entries of $A$ in the first $j$ row and columns of $A$. Suppose that $det(A_j ) \neq 0$ and $det(A_1) > 0$. Give and prove a rule in terms of the signs of the $det(A_j)$ to determine the signature of the Hermitian form defined by $A$.
I am unfamiliar with signatures of matrices, and after reading more about them I am still really unsure how to solve this problem. All I know is that we have very specific rules about the leading principal minors for the definiteness of the matrix, ie if all positive then all the eigenvalues are positive, negative definite if all the odd principal minors are negative, etc but have no idea how to derive a general rule to exactly determine the number of positive and negative eigenvalues.
I tried looking at the spectral decomposition and its respective principal minors, but I wasn't really getting anywhere with this, since the submatrix of $A = UDU^*$ seems not to be something I can easily simplify. I also tried looking into the min-max theorem and applying something like this: Eigenvalues of the principal submatrix of a Hermitian matrix but felt that this was not giving me very much information.
Also, since we know the characteristic polynomial splits with $n$ real roots, I thought maybe something like Descartes' rule of signs would be helpful like suggested here: Determining the signature of a Matrix based on the characteristic polynomial. However I was unsure how to find the coefficients of the determinant in terms of the given submatrix determinants, since we only know the leading principal minors and not all minors.
I am very lost and am unsure where to even start, and any help would be greatly appreciated.
