I have a symmetric real $(J+1) \times (J+1)$ matrix $H$, with structure
$$H=\begin{pmatrix} H_1 & H_2 \\ H_2^T & H_3 \end{pmatrix}$$
where $H_1$ is a symmetric $J \times J$ matrix, $H_2$ is a column vector with $J$ elements and $H_3$ is a scalar.
The elements of $H_1$ are given by $$H_{1(j,k)} = - \sum_{i=1}^n a_i b_i^2 x_{ij} x_{ik}$$ where $n$ is a positive integer, all $x_{ij} \geq 0$, all $a_i \geq 0$ with at least one $a_i > 0$, and all $b_i > 0$. Then from this answer, we know that if $X$ (the matrix of $x_{ij}$s, after removing rows corresponding to $a_i = 0$) has rank $J$, then $H_1$ is negative definite.
I hope to determine some conditions under which the larger matrix $H$ is also negative definite.
The elements of $H_2$ are $$H_{2(j)} = -c\sum_{i=1}^n b_ix_{ij}$$ where $c > 1$, and $$H_3 = -\sum_{i=1}^n c^2 r_i + d^2 y_i$$ where all $r_i > 0$, all $y_i \geq 0$, and $d > 1$.
I have started trying to prove that $v^THv < 0$ for any $v \neq 0$, i.e. $$v_{[J]}^T H_1 v_{[J]} + 2v_{J+1} (v_{[J]}^T H_2) + v_{J+1}^2 H_3 < 0$$ where $v_{[J]}$ denotes the subvector containing the first $J$ elements of $v$.
I know that the first component is negative if $X$ has rank $J$, and the last component is always negative, but I'm finding the middle part tricky.
Is there a property relating the definiteness of a principal submatrix to the definiteness of the larger matrix that I could use? Or is the approach I have started on the best one?
Thanks for any suggestions!