Interpolation of Matrix Weighted Spaces and Commutator Estimates

We present a formula for the interpolation of matrix weighted spaces of vector valued functions via interpolation functors. We apply our formula to the particular case of interpolation of matrix weighted $L^p$ spaces by the real and complex methods, and present consequences regarding the matrix Muckenhoupt classes and commutator estimates of Calder\'on-Zygmund operators with matrix $BMO$ functions. In particular, we characterize the logarithms of Muckenhoupt weights through higher order commutators.