What is the topology of the subspace $X_n\subset M_n(\mathbb R)$ consisting of all nonzero $n\times n$ matrices $A$ such that $A^t A$ is diagonal ? For example $X_2$ is the product of a torus and an interval. Is $X_n$ a manifold ? Reference sought.

1$\begingroup$ Isn't it just some sort of product of the space of orthogonal matrices with $\mathbb{R}^{n} \backslash \{0\}$ ? $\endgroup$ – Geoff Robinson Jun 19 '19 at 22:17

2$\begingroup$ That is true for $X_n\cap GL(n,\mathbb R)$ $\endgroup$ – Daryl Cooper Jun 19 '19 at 22:45

$\begingroup$ In that case you want diagonal matrices with all nonzero entries. I allow some ( but not all) zero entries in the diagonal matrix, but uniqueness of the product is admittedly lost when $A$ is singular. $\endgroup$ – Geoff Robinson Jun 20 '19 at 6:21

1$\begingroup$ I should have said above that $A$ has the form $CD$ where $C$ is orthogonal and $D$ is a nonzero diagonal matrix (this product is not unique, though, when $A$ is singular. Actually, I suppose that even when $A$ is invertible, we are allowed to change the signs of the diagonal entries of $D$, but I think we obtain some sort of uniqueness in the invertible case if we insist that the nonzero entries of $D$ are positive). $\endgroup$ – Geoff Robinson Jun 20 '19 at 8:30

$\begingroup$ Yes, so $X_n$ is the orbit in $M_n(\mathbb R)$ under the action of the orthogonal group of the set of nonzero diagonal matrices. Thus $X_n$ is stratified by rank. I am particularly interested in how these strata fit together. It is closely related to a moduli space problem for certain projective manifolds. $\endgroup$ – Daryl Cooper Jun 20 '19 at 15:01