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?


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.

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

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