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Traditional calibration methods based on scalar self- calibration rely on a linearized quasi-scalar approximation. Several underlying assumptions will no longer hold for the phased-array telescopes being considered in this conference. The reason is that the element antennas will show a strong instrumental polarization, that is moreover strongly dependent on the pointing of the array. Scalar selfcal cannot be used for calibrating such an array. This paper offers a rigorous description, based upon 2 X 2 matrices, that fully accounts for polarization phenomena in an interferometer. It then translates the traditional selfcal algorithm into this matrix language by exploiting the analogies between scalar and matrix multiplications. It is shown that the matrix algorithm does not yield a complete calibration: It only aligns all antenna- based error by suppressing the scattering of radiation away from source components to places in the image that should be empty. In doing so, it satisfies the requirement for a high dynamic range. However, it admits an unknown uniform in-place transformation, the poldistortion, of the matrix brightness. In terms of Stokes brightness parameters, the polvector (Q,U,V) in Stokes-vector space is rotated in an unknown way and there is an unknown mutual polconversion between Stokes I and the polvector. One must eliminate this poldistortion to make the image a faithful rendition of the source. Unpolarized sources can be used as calibrators to suppress the polconversion effect, after which prior statistical knowledge about the orientations and ellipticities of the antenna elements serves to eliminate most of the polrotation, much in the same way as in the quasi-scalar method. For homogeneous arrays of identical elements, one supplementary phase measurement of some sort is required; for heterogeneous arrays this is not even necessary. In the absence of unpolarized calibrators, polconversion must be eliminated by other means. There are no obvious ways of achieving this: Developing suitable calibration techniques will be a major challenge for the new generation of telescopes. In the concluding section, the consequences of my findings for the design of phased-array aperture-synthesis telescopes are explored.
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