Let $0 \leq x_{i-1} < x_i < x_{i+1} \leq 1$. Let $p, q$ be functions that depend on that such that $p$ is positive and $q$ is non-negative.
Let $c_i = a_{i+1,i} = a_{i,i+1}$.
Let all other coefficients be zero.
I try to prove that the matrix with coefficients shown below is positive definite. I've shown so far by the Gershgorin circle theorem that the eigenvalues are less than or equal to $|c_i| + |c_{i+1}|$ but I do not know how to show zero is not an eigenvalue of this matrix (equivalently, the matrix is invertible).
What theorems should I use? I cannot possibly row-reduce this $n$ by $n$ matrix, nor can I show it has $n$ pivot positions in this lifetime (it would be nice if I could)...
Diagonal coefficients are $$a_{i,i} = \left(\frac{1}{x_{i}-x_{i-1}}\right)^2\int_{x_{i-1}}^{x_{i}} [p + q(x - x_i)^2] dx \\ +\left(\frac{1}{x_{i+1}-x_i}\right)^2\int_{x_i}^{x_{i+1}} [p + q(x_{i+1} - x)^2] dx$$
Off-by-one-diagonal coefficients are $$c_i = \left(\frac{1}{x_{i+1}-x_i}\right)^2\int_{x_{i}}^{x_{i+1}} [-p + q(x - x_i)(x_{i+1} - x)] dx$$