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.

$$ A=D^{-1}MD\\ \frac{A_{max}}{A_{min}} < K, K>0 $$

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 can preserve the tridiagonal property is even better.

Thank you.

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.

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.

$$ A=D^{-1}MD\\ \frac{|A_{max}|}{|A_{min}|} < K $$

Thank you.

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.

$$ A=D^{-1}MD\\ \frac{A_{max}}{A_{min}} < K, K>0 $$

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 can preserve the tridiagonal property is even better.

Thank you.