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Since the pioneering work of S.-I. Amari (1977) and E. Oja (1982; 1989; 1992), principal component neural networks and their extensions have become an active adaptive signal processing research field. One of such extensions is minor component analysis (MCA), that proves to be effective in tasks such as robust curve/surface fitting and noise reduction. The aims of the paper are to give a detailed and homogeneous review of one-unit first minor/principal component analysis and to propose an application to robust constrained beamforming. In particular, after a careful presentation of first/minor component analysis algorithms based on a single adaptive neuron, along with relevant convergence/steady-state theorems, it is shown how the adaptive robust constrained beamforming theory by H. Cox et al. (see IEEE Trans. Acoust. Speech. Sig. Process., vol.34, no.3, p.393-8, 1986; vol.35, no.10, p.1365-76, 1987) may be advantageously recast into an MCA setting. Experimental results obtained with a triangular array of microphones introduced in a teleconference context help to assess the usefulness of the proposed theory.