I want to know what conditions are required to be able to identify a function from a manifold of matrices to another manifold of matrices (e.g. GL(n,R) to GL(n,R)) as being full-rank, so that I can apply (for example) the constant rank theorem. I understand that this involves vectorizing both the domain and codomain, but this typically produces very messy expressions, and it's not clear how to prove that it is full-rank from this.
In particular, suppose I have an expression for the derivative of a matrix function in the form of a matrix of matrices (see here), where the (i,j) entry is the derivative of the function with respect to the (i,j) element of the argument to the function. Intuitively it seems like I should be able to prove that the derivative is full-rank by proving that the rows of this matrix are linearly independent, where linear combinations of rows are done by multiplying each row by an arbitrary matrix rather than a scalar. Can anyone help me prove or disprove this, and is there a simpler way?