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Is it possible to say anything about the eigenvalues and eigenvectors of a matrix

$X = Y \circ xx^T$

where $Y$ is a positive definite symmetric matrix with known eigen-decomposition

$Y=U\Lambda U^T$

and $x$ is a vector. In particular if

$X=V\Phi V^T$

what is the relationship between $V,U$ and $\Lambda, \Phi$. And/or is there an $O(n^2)$ to compute $V,\Phi$ from $U,\Lambda$ where $n$ is the size of the matrices.

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  • $\begingroup$ Note that $X = D Y D^T$, where $D=D^T$ is the diagonal matrix with the entries of $x$ on the diagonal -- this may be a more useful way to think about the problem. (That said, I don't know the answer.) $\endgroup$
    – Federico Poloni
    Commented Jul 18, 2017 at 8:05
  • $\begingroup$ I had the same question come up - did you find an answer? $\endgroup$
    – smörkex
    Commented Feb 2, 2021 at 18:33
  • $\begingroup$ No not really. In the end I was able to recast my problem as K + zz.T and use the techniques mentioned in 8.4.3 of Golub math.ecnu.edu.cn/~jypan/Teaching/books/… $\endgroup$
    – Tzonathan
    Commented Feb 4, 2021 at 10:53
  • $\begingroup$ Thanks, Ill check it out $\endgroup$
    – smörkex
    Commented Feb 5, 2021 at 2:13

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