Possible methods to show that there is (existence or computing) a negative eigenvalue? I'm in a slight pickle. I'll be honest and state upfront that it's for my thesis, but I'm asking for methods, so I don't think it's cheating. I have a matrix $A$ over $\mathbb{R}$, symmetric, quite sparse, and I know from MatLab that it has a negative eigenvalue (relevant for my thesis). Now I would like to show that $A$ has a negative eigenvalue without having to reveal that I 'eig(A)'-ed the matrix (the function eig() gives you the eigenvalues in MatLab).
What I have tried:

*

*Gerschgorin circles won't help in this case.

*Inverse iteration (a modification of the power method). But this is very awkward as I will have to say something like, 'let $\mu = -k$' (the inverse iteration lets us try to find an eigenvalue closest to a $\mu$). Well how do I justify $\mu=-k$? Practically I have given away that I knew the eigenvalues in advance.

*Power method + rank deflation. True, this will work, and is currently what I'm opting for, but still kinda awkward in some sense.

*Determinant is negative, and my matrix is indeed of odd dimension! But, I would like to ask for other methods, since I love knowing more anyway...

*I'd like to avoid straight up computing $(A-\lambda I)v=0$, but interesting variations are ok.

I would like to know if there are other numerical/exact methods to do this? I'm ok if the method is a bit long/convoluted, I am eager to learn something new.
Edit: Basically I would like to know methods to show that a matrix has a negative eigenvalue without explicitly computing it.
 A: The standard tool for this situation is the LDLT factorization, a generalization of Cholesky's factorization to symmetric matrices that are not necessarily symmetric positive definite.
Given a symmetric matrix $A$ we can compute a unit lower triangular matrix $L$ and a diagonal matrix $D$ such that $$A = LDL^T.$$ By Sylvester's law of inertia, A and D have the same number of negative, zero and positive eigenvalues. This makes it trivial to determine the number of eigenvalues that are, say, negative.
By computing the LDLT factorization of, say, $A-aI$ and $A-bI$ we count the number of eigenvalues in the interval $[a,b]$. This is immensely useful when you wish to assert that you have found all eigenvalues in an interval.
Naturally, this assumes the use of exact arithmetic and it is possible to construct cases where the count is off because rounding errors changes the sign of an eigenvalue that is nearly zero.
The cost of computing the LDLT factorization of an $n$ by $n$ dense matrix is $O(n^3)$ arithmetic operation and $O(n^2)$ words of storage. You may be able to factor a sparse matrix using substantially fewer resources, but this is not certain and you may have to treat the sparse matrix as a dense matrix.
In LAPACK the function that computes LDLT factorization in IEEE double precision arithmetic is DSYTRF.
EDIT: In MATLAB the relevant function is ldl.
