Matrices seem to be the most common parametrization of linear transformations with the geometrical meaning of columns of the matrix being the re-mapped basis vectors. But sometimes it's difficult to visualize what exact effect will a linear transform (represented by a matrix) have on, let's say, the unit circle.
For example, I see eigenvectors and eigenvalues as an alternative parametrization of linear transformations, because they tell in which directions and by how much does a linear transformation stretch space, although some linear transformations have complex eigenvectors and eigenvalues, which breaks this stretching analogy for me. Another flaw in this analogy for me is that I fail to see why e.g. $2\times 2$ matrices cannot encode more than 2 axes of stretching (more than 2 eigenvectors).
When playing with 2D basis vectors, I saw that shear transforms produce results similar to camera movement in 3D space. Maybe N-dimensional transforms can be also expressed as certain operations on (N+1)-dimensional space?
Are there any other useful parametrizations of linear transformations that could perhaps bring more intuitive insight into how non-trivial (non-orthogonal) linear transformations affect space?