0
$\begingroup$

Given a diagonally dominant positive semi definite matrix $A \in R^{n\times n}$, with eigen decomposition $A = U\Sigma U^T$, can we say anything about the form of $U$.

For example, to sample uniformly from the space of matrices with eigenvalues as the diagonal elements of $\Sigma$ we sample the first vector of $U$ from the unit hyper sphere $U_1 \sim \text{uniform}(S^{D})$, and the $n^{th}$ vector, $U_n$, from the hypersphere spanned by the subspace orthogonal to $U_{1:n-1}$. Is this also the case for diagonally dominated matrices?

$\endgroup$
3
$\begingroup$

In the case $A = I$ (as diagonally dominant as you can get!), $U$ can be any orthogonal matrix, so in this case the answer is no.

There might be something you could say if the eigenvalues of $A$ are all distinct.

$\endgroup$
  • 1
    $\begingroup$ Very good point! Thank you $\endgroup$ – j__ Mar 4 '16 at 18:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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