I have a symmetric matrix $A$ of size $(n+1) \times (n+1)$ that I want to show is positive definite,
$$ A_{ij} = \begin{cases} 0 &\quad\text{if } i+j \text{ odd}\\ \frac{2}{i+j+1} &\quad\text{if } i+j \text{ even}\\ \end{cases}. $$
I know that $A$ is positive definite $\iff$ $\mathbf{x}^TA\mathbf{x}>0$ for all $\mathbf{x} \neq 0$.
I have attempted to solve this using summations:
$$ \mathbf{x}^TA\mathbf{x} = \sum^n_{i=0}x_i\sum^n_{j=0}A_{i,j}x_j . $$
However when I try to simplify this nothing seems to quite work, eg.
$$ \mathbf{x}^TA\mathbf{x} = \sum^n_{i=0}x_i\sum^{n/2}_{j=0}A_{i,2j}x_{2j} = \sum^n_{i=0}x_i\sum^{n/2}_{j=0}\frac{2}{i+2j+1}x_{2j} , $$ clearly this is not true.
I'd really appreciate a nudge in the correct direction.
Thanks.