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Assume that $A\in\mathbb{R}^{n\times n}$ is a symmetric positive definite matrix and $D=diag(d_1,\ldots,d_n)$ is a diagonal matrix constructing by using the diagonal entries of $A$, indeed $d_i=a_{ii}$.

What can we say about the relationship between the eigenvalues and eigenvectors of $A$ and the eigenvalues and eigenvectors of $D^{-1/2}AD^{-1/2}$?

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There is not much to say.

$D^{-1/2} A D^{-1/2}$ is positive definite. It has diagonal entries $1$, so the sum of its eigenvalues is its trace, which is $n$. The trace of $A$ could be any positive number.

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  • $\begingroup$ I think you are right. Thank you $\endgroup$ – like_math Dec 26 '18 at 22:00

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