The group $\mathrm{GL}_n(\mathbb{R})$ inherits a metric under the injection $$\mathrm{GL}_n(\mathbb{R}) \hookrightarrow \mathbb{R}^{n^2}$$.
Is there a natural/geometric way of viewing this metric on $\mathrm{GL}_n(\mathbb{R})$? What does is mean for two linear transformations to be close? What properties do two "close" linear transformations share?
Viewing it as a norm on $n^2$-dimensional Euclidean space (rather than restricting to the general linear group), it is called the Frobenius norm (or inner product). For matrices with good spectral properties (and hence I think for all matrices), it coincides with the Schatten norm, which is the diagonal length of image of the unit box under multiplication by the matrix. Compare that with the operator norm, which is the length of the longest side of the image of the unit box.
See the excellent answer by Eric Naslund here.