This paper shows that there are intimate connections between the quadratic transform technique for solving fractional programming (FP) problems and the Schur-complement technique in matrix analysis. We demonstrate that the quadratic transform technique is related to two aspects of the Schur complement: (i) the linear matrix inequality (LMI) condition for positive semidefiniteness and (ii) the matrix determinant formula. Specifically, we establish that the quadratic transform and the Schur-complement LMI condition imply each other. This connection allows us to provide new interpretations of the auxiliary variable in the quadratic transform, and it allows us to rederive the Schur-complement determinant formula. Furthermore, this connection leads to generalizations of the quadratic transform in FP and the Schur-complement LMI that can accommodate generalized matrix inverse. As an application in information theory, we apply the generalized FP framework to the least-favorable-noise minimax formulation of the Gaussian vector broadcast channel sum capacity problem. When the least-favorable noise covariance is singular, matrix-inverse-based Karush-Kuhn-Tucker (KKT) analysis would require a careful analysis of the input and output spaces of the channel. We show using generalized FP that an auxiliary-variable representation of the singular matrix fraction directly yields the reciprocal multiple-access channel and recovers the uplink-downlink duality relation for sum capacity.

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