### Complimentary eigenvalue of a matrix

How to obtain all the complimentary eigenvalues (also associated complimentary eigenvectors) of a given matrix. Complimentary eigenvalues and eigenvectors of a given matrix $A$ of order $n\times n$ is the solution to the following system

$x≥0_n$,
~~$Ax−λx≥0_n$,
~~$Ax−λx≥0_n$ and
$⟨x, Ax−λx⟩=0$ where ~~$x\in ~~$x(\neq 0)\in R^n$