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$\circ$ 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.

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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.

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  • $\begingroup$ Not quite automatically, if there is multiplicity. $\endgroup$
    – Igor Rivin
    Commented Nov 10, 2013 at 17:49
  • $\begingroup$ true, but any sensible (numerical) routine will return orthonormal eigenvectors even if there are degenerate eigenvalues. $\endgroup$
    – Carlo Beenakker
    Commented Nov 11, 2013 at 1:19

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