Exercise 7, § 3

a) Show that, in the notation of no. 5, the space $L_{n,p}$ is homeomorphic to the product of $V_{n,p}$ and $\mathbf{R}^{p(p+1)/2}$. Note that every matrix $X \in L_{n,p}$ can be expressed uniquely in the form $U . Y$, where $y \in V_{n,p}$ and $U = (u_{ij})$ is such that $u_{ij} = 0$ whenever $i < j$, and $u_{ii} > 0$ for $1 \leqslant i \leqslant p$. Put $Y = f(X)$.

b) In the notation of no. 5, Proposition 8, show that $f$ is a homeomorphism of $B_\sigma$ onto $f(B_\sigma)$ [note that $X \to f(X^{-1}_\sigma X)$ is a continuous mapping of $A_\sigma$ onto $f(B_\sigma)$ and that $f(X^{-1}_\sigma X)$ belongs to the same class as $X$ mod $\Delta_{n,p}$; then apply Lemma 2]. Deduce that the intersection $D_\sigma$ of $A_\sigma$ and $V_{n,p}$ is homeomorphic to the product of $V_{p,p}$ and $\mathbf{R}^{p(n-p)}$ [every matrix belonging to $D_\sigma$ is uniquely expressible as the product of a matrix of $V_{p,p}$ and a matrix of $f(B_\sigma)$].