Complimentary eigenvalue of a matrix

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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

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