We study matrix weights defined on the multivariate torus \(\mathbb {T}^d\) . We formulate the theory of matrix weights in the context of general coverings and in connection with the corresponding averaging operators. On this level of generality we prove the diagonalization theorem of Bloom and a form of its converse for the full range of Lipschitz functions. We incorporate within our setting the studies of log-preserving matrix weights, the BMO distance theorem for matrix weights, and the almost diagonal matrix weights.