Wikipedia and other sources say that the index of a non-degenerate critical point $p$ of a manifold $M$ is "the dimension of the largest subspace of $T_p\left(M\right)$ on which the Hessian is negative definite."
I know what it means for a matrix to be negative definite, but am not sure how this is relevant. Certainly, the Hessian of $f$ at the point $p$ is either negative definite or it isn't. Does Hessian here mean, e.g., for $f(x, y) = x^3 - y^3$,
$$\begin{pmatrix}6x & 0 \\ 0 & -6y\end{pmatrix}$$ as a function of $x$ and $y$? If so, it's not clear why this matrix why the set of $\left(x, y\right)$ for which this matrix is negative definite should form a subspace. To find the eigenvalues of this matrix (call it $H$), I must find the roots of $\det\left(H - \lambda\cdot I\right) = 0$. But $\det\left(A + B\right)\neq \det A\:+\:\det B$ in general, so that it's not clear why the matrix is negative definite for $\left(x_1 + x_2, y_1 + y_2\right)$ just because it's so for $\left(x_1, y_1\right) + \left(x_2, y_2\right)$.