# Jacobian of the action of a matrix on a Grassmannian

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?

• It might help to clarify a bit what you mean by "these types of calculations". In what level of generality do you want to work? For $G/P$ a flag manifold of a reductive group $G$, the action of $P$ on the tangent space at a point is equal to its action on the Lie algebra quotient $\mathfrak{g}/\mathfrak{p}$. The Jacobian may be computed as the determinant of this action. Is that the kind of direction you're thinking of going? – Will Sawin Dec 28 '18 at 0:55