We want to estimate the vectors $\{\boldsymbol{x,y}\}\in\mathbf{R}^{n}$ in a generalized solution to a system of linear equations.
$$ \tag{1} \boldsymbol{y}=A^+\boldsymbol{b}+(I_{n}-A^+A)\boldsymbol{x} $$
where $A\in\mathbf{R}^{m\times n}$ ,which can be inversible as Moore–Penrose pseudoinverse $A^+$, and $\boldsymbol{b}\in\mathbf{R}^{m}$ are defined. Furthermore, $\boldsymbol{y}$ must be satisfied with $\|\boldsymbol{y}\|=1$ and $\boldsymbol{y}\geq\mathbf{0}$.
As my attempt, we introduce an eigenvalue $\lambda$ and an eigenvector $\boldsymbol{x=p}$ in the matrix $(I_{n}-A^+A)$, and replace with equation $(1)$ as follows:
$$ \tag{1'} \boldsymbol{y}=A^+\boldsymbol{b}+\lambda\boldsymbol{p} $$
However, in these case, it cannot be ensured $\boldsymbol{y}$ is any positive.