I have an equation as below,
$$Rw = \lambda_1R_aw + \lambda_2R_bw $$
where, $R$, $R_a$, and $R_b$ are positive definite at least semi-positive definite and Hermitian matrix. $\lambda_1$ and $\lambda_2$ are getting from Langrage multipliers. We consider this equation in complex domain.
$R$, $R_a$, and $R_b$ are $M \times M$ matrix, respectively. $w$ is $M \times 1$ vector.
At this moment, can we get the general solution like a single Lagrange multiplier case? In other words, only one Lagrange multiplier case considers as Eigenvalue problem i.e., $ Rw = \lambda w .$
If we need to get the closed form solution or numerical soultion from the above equation, do we need more information?
Although I found the articles concerning two parameter eigenvalue problem, it consists with two equations. However in this case, we get only one equation with two eigenvalues(or Lagrange multipliers).
Because the number of unknowns(two) is bigger than the number of equations(one), is there no general solution?
Thank you for reading this question.
You actually have $M$ conditions. Even if the equation is just one , it has to be sattisfied in all $M$ coordinates. Your uknowns are $M+2$ , one for each coordinate plus $\lambda_1$ $\lambda_2$. But remember that the solution of a system of non linear eqations is not necesserely unique ( I'm sorry but i don't know any theorems on the solvability of non linear equations).