Gaussian Processes are ill-conditioned (Example)

Someone asked me "why did you say in the Gaussian processes tutorial that Gaussian processes are ill-conditioned?" Here is an answer in the form of a tiny exercise.

Take the simplest case where the function is just a constant.

y(x) = Y for all x. (Y is Gaussian distributed and not known a priori)

Make a GP with the appropriate covariance function.

Now

(1) think about the inference of the GP given data {x,t}. (Not very difficult is it?)

(2) see what happens when you use the standard GP matrix inversion formalism to solve the problem. What is the well-conditioned-ness of the matrix you must invert?