In covariance matrix estimation, one of the challenges lies in finding a suitable model and an efficient estimation method. Two commonly used modelling approaches in the literature involve imposing linear restrictions on the covariance matrix or its inverse. Another approach considers linear restrictions on the matrix logarithm of the covariance matrix. In this paper, we present a general framework for linear restrictions on different transformations of the covariance matrix, including the mentioned examples. Our proposed estimation method solves a convex problem and yields an $M$-estimator, allowing for relatively straightforward asymptotic (in general) and finite sample analysis (in the Gaussian case). In particular, we recover standard $\sqrt{n/d}$ rates, where $d$ is the dimension of the underlying model. Our geometric insights allow to extend various recent results in covariance matrix modelling. This includes providing unrestricted parametrizations of the space of correlation matrices, which is alternative to a recent result utilizing the matrix logarithm.