We develop Bayesian learning methods for low-rank matrix reconstruction and completion from linear measurements. For under-determined systems, the developed methods reconstruct low-rank matrices when neither the rank nor the noise power is known a priori. We derive relations between the proposed Bayesian models and low-rank promoting penalty functions. The relations justify the use of Kronecker structured covariance matrices in a Gaussian-based prior. In the methods, we use expectation maximization to learn the model parameters. The performance of the methods is evaluated through extensive numerical simulations on synthetic and real data.