Let $A=(a_{i,j})$ be a $n$ x $n$ matrix $(n\geq 2)$, where $a_{i,i} = |x|^2-2x_i ^2$ and $a_{i,j} = -2x_i x_j$ for $i\neq j$. Here $|x|^2 = x_1^2+x_2^2+ \cdots + x_n^2$. I'd like to compute the determinant of $A$.
Calculations suggest that $\det A = -|x|^{2n}$ and it is true for $n=2,3,4$. But I have no idea how to prove/disprove that identity for general $n$.
This matrix came up when computing the Hessian of the function $\phi(\xi) = \log |\xi| + \frac{1}{\tau}x\cdot \xi$ at its critical point $\displaystyle \xi_0 = -\frac{\tau}{|x|^2}x$. I was estimating an oscillatory integral by using the stationary phase method.