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A matrix description of microwave amplifiers such as klystrons, traveling-wave tubes, and backward-wave amplifiers, in which an electron beam interacts with longitudinal RF fields, is developed. Certain relations between the matrix elements are derived as a consequence of the conservation of energy and these relations set a lower limit to the noise figure attainable with amplifiers of this class. It is shown that the minimum noise figure of any amplifier of this type with lossless RF structures is identical with that already found by several authors for the traveling-wave tube and is entirely determined by the noise parameters of the beam. These in turn depend only on conditions in the immediate neighborhood of the cathode. Special cases involving lossy structures are investigated and in each case the presence of loss is shown to increase the noise figure. The method is also applied to calculate the minimum noise figure of a double-stream amplifier.