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Define a set $A = \{x_i/x_i\in\mathbb{R}^m, i = 1,2,3..n\}$. Let $f:\mathbb{R}^m\to(0,\infty)$ be an even symmetric positive definite function.

Let $D = [d_{i,j}]$ be an $n\times n$ matrix such that $d_{i,j} = f(x_i-x_j)$

Let $\epsilon = \min\limits_{i,j} \|x_i-x_j\|_2$ and assume $\epsilon > 0$.

Consider the matrix $D+\alpha I$, where $I$ is an identity matrix and $\alpha>0$. Naturally $D$ is a positive semi definte matrix as the function $f$ is a positive definite function. So $D+\alpha I$ is positive definite.

I am looking for an upper bound on the condition number of the matrix $D+\alpha I$ in terms of $\alpha$, $\epsilon$ and the function $f$. Does such a bound exist?

Condition number defined as the ratio of magnitude of largest eigenvalue to the magnitude of the least eigenvalue.

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  • $\begingroup$ "$D$ is positive semi-definite" is a false assertion. Its diagonal is actually made of zeros ! $\endgroup$ – Denis Serre Jun 24 '20 at 5:45
  • $\begingroup$ @DenisSerre : Sorry its a mistake. Editing the question. $\endgroup$ – Rajesh D Jun 24 '20 at 5:57
  • $\begingroup$ @DenisSerre : Edited. Thanks for pointing it out. Hope its good now. $\endgroup$ – Rajesh D Jun 24 '20 at 6:05

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