How bad are symmetric Pick matrices?
Dario Fasino, Vadim Olshevsky · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2000
Let P be a symmetric positive definite Pick matrix of order n. The following facts will be proven here: (1) P is the Gram matrix of a set of rational functions, with respect to an inner product defined in terms of a 'generating function' associated to P; (2) Its condition number is lower-bounded by a function growing exponentially in n. (3) P can be effectively preconditioned by the Pick matrix generated by the same nodes and a constant function.