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Let $x,y\in R^d$ and $d:R^d\times R^d \rightarrow R$ a metric on $R^d$ be given.

The exponential kernel is defined by:

$k(x,x')=e^{−αd(x,x')}$ where $α>0$.

The kernel matrix is defined as the Gram matrix of $k$: $K_{ij}=k(x_i,x_j), i,j∈[1…n]$.

Is it possible to prove that $K$ is a positive (semi-positive) definite matrix, i.e. $k$ is a positive definite Mercer kernel?

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