I'm working on a problem whose solution relies on finding if an arbitrary matrix A is positive semi-definite. A is real valued, square and symmetric, and each of its entries are given by:
$A_{ii} = f(i)$
$A_{ij} = A_{ji} = f(i,j)$
The problem is that this is part of an induction proof, where the size N of the matrix represents the problem at step N. Are there any simple ways of finding out if A is positive semi-definite? Finding the characteristic equation is pretty hard, given that it would rely on finding the determinant of an arbitrarily large matrix A(NxN).
Thank you.
EDIT: I'm looking for a method that is non dependent on f(), since my f() is quite extensive. Some analogous to finding the characteristic equation using the simple:
$det(A-\alpha I) = (f(1) - \alpha)(f(2)-\alpha) - f(1,2)^2$
for N=2