Let $V$ be the vector space of all $2$ by $2$ matrices. Define $\langle A, B\rangle = \mathrm{tr}(A^TDB)$, where $D = \begin{pmatrix}7&1\\ 1&1\end{pmatrix}$. Prove that $<A, B>$ defines an inner product on $V$.
So I do see that the inner product is symmetric and bilinear as $D$ is equal to its transpose. But I struggling to prove the idea of positive definiteness as that would require that $D$ have positive eigenvalues.
But after calculating the eigenvalues, I get that they are $4+\sqrt{10}$ and $4-\sqrt{10}$. Since the second eigenvalue is negative, I am not sure how this inner product is positive definite.
Any help would be highly appreciated!