I've been studying kernel methods in my course and in a textbook came across an additional line (aside from the positive definiteness of the Gram/Kernel Matrix).
That it should be enough for a kernel to be a valid inner product.
Now for one of last years exercises I found that $\frac{1}{K(x,y)}$ for a valid Kernel $K(x,y)$ is not a valid kernel. The proof is given by showing that the matrix is not necessarily positive definite (which is easy to follow). That however confuses me, because if the rule stated above holds, then $K(x,y)$ is an inner product. I can show that the reciprocal is also symmetric and linear with a little algebra. Positive semi-definitenes seems to also be guaranteed. And thus I arrive at contradictory statements. Do I overlook something major (probably)?
I am suspecting I am doing something wrong with my attempt at showing the reciprocal is an inner product.
thank you!
To summarize the discussion in the comments: the main misconception seems to have been that if $K(x,y)$ is linear in each of its arguments, then $1/K(x,y)$ must also be linear in each of its arguments. However, this is not the case. Thus, the condition that $1/K(x,x) > 0$ for all $x$ is not sufficient to ensure that $1/K(x,y)$ defines an inner product.