Cayleyʹs theorem and its application in the theory of vector Padé approximants

Le A be a matrix of even dimension which is anti-symmetric after deletion of its rth row and column and let R, C be the anti-symmetric matrices formed by modifying the rth row and column of A, respectively. In this case, Cayleyʹs (1857) theorem states that det A = Pf R · Pf C, where Pf R denotes the Pfaffian of R. A consequence of this theorem is an explicit factorisation of the standard determinantal representation of the denominator polynomial of a vector Padé approximant. We give a succinct, modern proof of Cayleyʹs theorem. Then we prove a novel vector inequality arising from investigation of one such Pfaffian, and conjecture that all such Pfaffians are nonnegative.