mmWave antenna array calibration is a necessary yet tedious and costly process in manufacturing to capture the non-idealities in phased arrays, in order to obtain codebooks for accurate and stable beam steering. Unfortunately, predefined codebooks provided by manufacturers to steer beams in a given set of directions do not support the arbitrary beam shapes required for various mmWave communication, sensing, and security applications. To create arbitrary beam patterns, one needs to first find the unknown calibration vector for the particular phased array in use. In this paper, we introduce EiCal, a novel zero-shot technique that leverages the beamforming codebook advertised by the manufacturer to extract the calibration vector at zero cost (i.e., with no additional measurements). The key idea is that the unknown desired calibration vector can be obtained via an appropriately designed eigen-decomposition of the given codebook. We experimentally demonstrate the efficacy of EiCal on a 60 GHz mmWave array for two scenarios: angle estimation using compressive pseudorandom beams, and simultaneous steering of beams and nulls. Our results also point to potential simplifications in the calibration process at the manufacturer.