Eigenvalues and index If I have a set of eigenvalues for a Hessian matrix of a non-degenerate critical point, and I have to determine its index, must I consider all negative eigenvalues, or must the negative eigenvalues be unique? I mean, $$(\lambda + 1)(\lambda + 2)^2 = 0 $$ is the determinant of the diagonalized Hessian of a function with three degrees of freedom. The values are $\lambda =-1, \lambda =-2, \lambda =-2$. Is the index $3$ or $2$? Alternatively, is the critical point a local maximum or a saddle point? 

The eigenvalues don't have to be unique, but you have to take into account their multiplicities, or what is nearly the same thing, the dimension of the entire eigenspace corresponding to each eigenvalue, to make sure you obtain/account for every direction in which the given function is decreasing.  Recall that a Hessian matrix at a critical point is symmetric; thus the peculiarities associated with generalized eigenvectors etc. do not occur in this situation; the dimension of an eigenspace is the multiplicity of its eigenvalue.   In your problem, the index is $3$.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
