# Bounding the condition number of a matrix associated with an even symmetric positive definite function

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.

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