$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$ 
where matrice $A \in \mathbb{R}\_{n \times n}$ is a positive semi-definite with eigenvectors $x = (x\_{1},x\_{2},....,x\_{n})\in \mathbb{R}\_{n \times n}$ with $x\_i=(x\_i(1),x\_i(2),...,x\_i(n))^{T}$ and eigenvalues $ \lambda = (\lambda\_{1},\lambda\_{2},...,\lambda\_{n}) \in \mathbb{R\_{+}}$.


$\circ$ What kind of method can be used to solve (1) so as to determine $x = (x\_1,x\_2,....,x\_n)$ such that $$\displaystyle{\sum\_{i=1}^{n}x\_{i}^{2}(1)=\sum\_{i=1}^{n}x\_{i}^{2}(n)}$$


Thank you.



Eigenvalue problem with quadratic constraints
since you say that $A$ is positive semi-definite, you're restricting yourself to real symmetric matrices $A$, so the matrix $x$ of eigenvectors is an $n\times n$ orthogonal matrix, with $\sum\_{i=1}^{n}x\_{i}^{2}(j)=1$ for all $j=1,2,\ldots n$. --- your constraint is therefore satisfied automatically.


