Let $A=(a\_{ij})\_{i,j=1}^n$ be a symmetric real matrix, $M\_k:=det(a\_{ij})\_{1\leq i,j\leq k}$ be its minors and $M\_k\ne 0$ for all $k$. Then signs of eigenvalues of $A$ are equal (up to some permutation) to signs of $M\_1$, $M\_2/M\_1$, $\dots$, $M\_{n}/M\_{n-1}$. It is clear by induction, for example: when we replace $n-1$ to $n$ by adding last row and last column, we either add one positive eigenvalue or add one negative (number of, say, positive eigenvalues may not decrease by variational principle). The sign may be obtained by the sign of product of all eigenvalues, which equals to $M\_n$.


What I ask is the reference to this easy, but somehow useful statement. I completely agree that it is not quite of research level and so appreciate its possible closing. 



signs of eigenvalues of quadratic form
This is very closely related to Sylvester's law of inertia (see <http://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia>).


