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An RF beam-forming matrix is described which forms ``n'' simultaneous beams from an ``n'' element array in a passive and theoretically lossless manner. The principle of operation is explained using some simple matrix configuration. A general expression for the far-field pattern of any beam is derived and then used to study the positions of beam peaks, the position of beam nulls, the crossover level between beams and the frequency sensitivity of beam positions. This matrix provides a uniform illumination of the array aperture; however, simple beam combining techniques will yield tapered illuminations. An experimental 16-element beam-forming matrix which operates at 900 Mcps is described, and results of RF and antenna measurements on the matrix are presented.