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The minimal realizations of pseudo-positive, pseudo-bounded, and pseudo-Schur rational matrix functions are constrained to satisfy certain matrix inequalities involving a Hermitian matrix. We show that the congruence class of this Hermitian matrix is determined by the Oono-Imamura index of the pseudo-positive matrix function. In the pseudo-lossless case, the connection with the matrix Cauchy index is established. A remarkable relationship between the realizations of a pseudo-positive matrix and the corresponding pseudo-Schur matrix is pointed out.