The group of unimodular matrices $\mathbb{U}[s]^{n\times n}$ is given by the set of $n\times n$ square (real) matrix-valued polynomials $\mathbb{R}[s]^{n\times n}$ which admit a polynomial inverse. Equivalently, $\mathbb{U}[s]^{n\times n}$ coincides with the group of $n\times n$ square (real) matrix-valued polynomials that have non-zero constant determinant. My question is the following one: does there exist a "natural" metric on the space $\mathbb{U}[s]^{n\times n}$ which, in some sense, resembles to the "natural", i.e. affine-invariant, metric of the general linear group $\mathbb{GL}(n)$ for the case of $n\times n$ constant matrices?
Any reference/comment is very appreciated. Thanks.
