Theorems of alternatives on the feasibility of linear matrix inequalities (LMIs) are used in order to provide novel simple proofs for two considered versions of the Kalman-Yakubovich-Popov (KYP) lemma for discrete-time positive linear systems. Two different and novel recursive methods, to determine whether a positive matrix is or is not Schur, are obtained as an application of an existing connection between the strict inequality version of the KYP lemma for single-input single-output (SISO) discrete-time positive linear systems and a Schur matrix condition. Examples are included which provide illustration on these recursive methods.