🚀 go-pugleaf

>My question(s) are: is this really true? How does one set about
>proving (or disproving) positive-definiteness? what is the
>physical significance of positive-definiteness?

The existence of a definite coefficient matrix is a sign that the 
operator form of the problem can be obtained by an extremum principle
(minimum principle => positive definite, max => neg def). This is
typically possible for self-adjoint, (positive) definite operators. 
In this case, the Ritz method and the Galerkin method are equivalent, 
and it can be shown that the resulting approximation is optimal, in the 
sense that it minimizes the discretization error (in an "energy" norm). 
It is easy to show that the coefficient matrix must be (positive) 
definite. The classic example is linear elasticity; the minimum 
principle is minimum potential energy or minimum complimentary energy. 
For example, the differential equation for axial deformation of an 
elastic bar -ku" = f can be obtained by finding u(x) that minimizes 
the functional \int 0.5k(u')^2 - fu dx. On the other hand, the 
convection-diffusion equation -ku" + cu' = f is not derivable by 
extremizing a functional, so we cannot expect the coefficient matrix 
be definite. 

Numerically, many nice properties accrue from positive definiteness. 
With direct methods, we need not worry about stability and are free to 
pivot to reduce fill-in. Certain iterative methods (e.g. Jacobi, Gauss 
Seidel, conjugate gradients) are guaranteed to converge from any 
starting iterate.