1
$\begingroup$

$\circ$ 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.

$\endgroup$

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Not quite automatically, if there is multiplicity. $\endgroup$
    – Igor Rivin
    Nov 10, 2013 at 17:49
  • $\begingroup$ true, but any sensible (numerical) routine will return orthonormal eigenvectors even if there are degenerate eigenvalues. $\endgroup$ Nov 11, 2013 at 1:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.