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We present new beampattern synthesis approaches based on semidefinite relaxation (SDR) for signal power estimation. The conventional approaches use weight vectors at the array output for beampattern synthesis, which we refer to as the vector approaches (VA). Instead of this, we use weight matrices at the array output, which leads to matrix approaches (MA). We consider several versions of MA, including a (data) adaptive MA (AMA), as well as several data-independent MA designs. For all of these MA designs, globally optimal solutions can be determined efficiently due to the convex optimization formulations obtained by SDR. Numerical examples as well as theoretical evidence are presented to show that the optimal weight matrix obtained via SDR has few dominant eigenvalues, and often only one. When the number of dominant eigenvalues of the optimal weight matrix is equal to one, MA reduces to VA, and the main advantage offered by SDR in this case is to determine the globally optimal solution efficiently. Moreover, we show that the AMA allows for strict control of main-beam shape and peak sidelobe level while retaining the capability of adaptively nulling strong interferences and jammers. Numerical examples are also used to demonstrate that better beampattern designs can be achieved via the data-independent MA than via its VA counterpart.