# Can this simple quadratic optimization problem be turned into a simple eigenvalue problem?

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 $$s$$.

1. Is there a closed-form solution if one knows the eigenvectors/values of $$A$$? I tried this but I keep going in circles...

2. 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 $$B$$?

(The optimality conditions of your problem, $$Ax+b+2\lambda x = 0 \\ x^Tx = 1$$ constitute an inhomogenous eigenvalue problem)