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Let $M$ be a positive $(M_{ij}>0 \ \forall i,j)$ symmetric matrix.

I wonder if there exists a similarity transformation $D$ that can control its maximum entry and minimum entry in a certain range or ratio? I.e. $$ A=D^{-1}MD\\ \frac{A_{\max}}{A_{\min}} < K, \quad K>0 $$

This problem comes from an algorithm I am currently working on, which is to solve for the eigenpair of $M$.

But the algorithm shows numerical instability when the above quotient is large. So I wonder if there is a similarity transformation that can control the quotient.

A transformation that can preserve the tridiagonal property is even better.

Thank you.

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  • $\begingroup$ You can transform your matrix $M$ into a diagonal matrix with positive entries. But the quotient can become arbitrary large. I don't understand your intention. $\endgroup$
    – Dieter Kadelka
    Commented Jun 17, 2021 at 12:06
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    $\begingroup$ Actually I am working on an algorithm to solve for eigenvalue and $M$ is actually tridiagonal. And I noticed that a large quotient will cause numerical instability in that algorithm. So I wonder if there is a similarity transform that can address this problem. $\endgroup$
    – 卢弘毅
    Commented Jun 17, 2021 at 12:43
  • $\begingroup$ The last comment clarifies the question a lot, perhaps add it to the 'main' body of the question? $\endgroup$
    – Vincent
    Commented Jun 17, 2021 at 13:15
  • $\begingroup$ Added, thanks, Vincent. $\endgroup$
    – 卢弘毅
    Commented Jun 17, 2021 at 14:37

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