I'm looking for a reference concerning a calculation found in Furstenberg's 1963 paper "Non-commuting random products".

Lemma 8.8 of this paper states that if one takes a $d\times d$ invertible real matrix $A$, then the Jacobian of its action on the space $(d-1)$-dimensional subspaces at a subspace $S$ of $\mathbb{R}^d$ is given by

$J_A(S) = |det(A)|^{d-1}/|det_S(A)|^d$,

where $det_S(A)$ is the determinant of $A$ restricted to the subspace $S$ (and one uses the inner-product from the ambient space to define the $(d-1)$-dimensional volume on $S$ and its image under $A$).

For example if $d = 2$ and $S$ is a 1-dimensional subspace of $\mathbb{R}^2$ genereted by a unit vector $v$ then

$J_A(S) = |det(A)|/\|Av\|^2$.

The proof is left to the reader.

While, I am able to reproduce the result, **I'm looking for a reference where these types of calculations are treated systematically**.

Does anyone know of such a thing?

Thanks in advance!