Expanding Quadratic form

Question

I am currently working on a problem and I am trying to convince myself that given a sequence of real number $\{s_n\}$, that truly the constants $c_i \in \mathbb{R}$ are zero if the quadratic form $$\sum_{i,j=0}^{n}s_{i+j}c_ic_j = 0.$$

I transformed the quadratic form into matrix form $c^TAc =0$ but still can not convince myself. Thanks

Answer 1

We have that matrix $A = [s_i + s_j]$ is a real symmetric matrix. Now, it doesn't have to be that $c^T A c = 0 \implies c = \mathbf 0.$ Consider the sequence $\{-1,0,1\}$ with the corresponding matrix $$A = \begin{bmatrix} -2 & -1 & 0\\ -1 & 0 & 1\\ 0 & 1 & 2 \end{bmatrix} .$$

and the vector $c = [1\quad 0\quad -1]^T$. It holds that $c^TAc = 0.$