  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.
