Let $\langle u, v\rangle = u^* \begin{bmatrix} 8 & -1 \\ -1 & 8 \end{bmatrix}$ $v$ be defined on $\mathbb{C}^2\times\Bbb C^2$. Prove, or disprove, that $\langle u, v\rangle$ is an inner product on $\Bbb C^2$.
According to my textbook (A First Course in Wavelets with Fourier Analysis, 2nd ed.), to prove something is an inner product, you need to prove positivity, conjugate symmetry, homogeneity, and additivity. I'm a bit stuck on the positivity part; I got $\langle u, u\rangle$ up to:
$$8|u_1|^2 - \bar{u}_1 u_2 + u_1 \bar{u}_2 - 8|u_2|^2.$$
Where should I go from here to show that this sum is greater than $0$? Thank you.