I'm interested in a type of problem on this form
$$\min_{x} x^{T}Ax+x^{T}b \quad \text{s.t} \quad x^{T}x=1 $$
where $A$ is positive definite. As you can see, if it weren't for the $x^{T}b$ term in the objective, this would be equivalent to finding the eigenvector with the smallest eigenvalue of $A$
This problem becomes trivial when $A=I$, where the solution will be on the form $x=s b$, with some scalar $s$.
Is there a closed-form solution if one knows the eigenvectors/values of $A$? I tried this but I keep going in circles...
Can this type of problem be transformed based on $A$ and $b$ so that the solution $x$ becomes an eigenvector/value of some derived matrix $B$?