How can we parametrise this matricial hypersphere?

What I call a matricial hypersphere for lack of a recognised name is the set in $\mathbb{R}^{p\times k}$ defined by
$$\mathfrak{H}=\left\{
a_1,\ldots,a_k\in \mathbb{R}^{p};\ \sum_{i=1}^k a_i a_i^\text{T} = \mathbf{A}
\right\}$$
where $\mathbf{A}$ is a $p\times p$ symmetric positive semi-definite matrix of rank $k$ $(k\le p)$. My questions are

  1. Is this a well-known object?
  2. Given the matrix $\mathbf{A}$ is there a completion of $\mathbf{A}$ into an object in bijection with $\{a_1,\ldots,a_k\}$, which is my meaning of parameterisation?
  3. what is the size or dimension of $\mathfrak{H}$?

Note: This object does not stem out of nowhere. It appears in linear
regression, where the $a_i$ vector is a collection of regression
coefficients, and in connection with Wishart distributions, where the
$a_i$’s are Normal variates. I actually need to find a
reparameterisation of the $a_i$’s given $A$ to proceed a research
problem.

Solutions Collecting From Web of "How can we parametrise this matricial hypersphere?"

Point 1) Let $B$ be the matrix with columns $a_i$: your description is equivalent to

$$\tag{1}BB^T=A$$

Thus, being given a symmetrical semi-definite positive $n \times n$ matrix $A$ with rank $k$, $\frak{H}$ can be identified with the set of $n \times k$ matrices $B$ such that $A$ can be written under the form (1).

Remark: formula (1) is “up to the multiplication by a $k \times k$ orthogonal matrix $\Omega$” (with property $\Omega\Omega^T=I_k$). More precisely, any decomposition of the form (1) generates a family of decompositions:

$$\tag{2}B\Omega\Omega^TB^T=A \ \ \Leftrightarrow \ \ B’B’^T=A \ \ \text{with} \ \ B’:=B\Omega$$

Point 2): Concerning parameterization, couldn’t you use the more or less classical parametrizations of the (grassmannian) manifold of $k$-dimensional subspaces in $\mathbb{R^n}$ ? A reference (http://www.macs.hw.ac.uk/~simonm/schubertcalculusreview.pdf). Let us take an example with $n=3$ and $k=2$ :

$$B^T=\begin{pmatrix}1&0&x\\0&1&y\end{pmatrix}$$

(I have taken $B^T$ because the “landscape” form is easier to work with).

The idea behind this parameterization which places into evidence a first block $I_k$ is this one :

Consider $B^T$, which is rank-$k$ matrix with $k$ rows and $n$ columns.

We can write it under the block form $B^T=(C|D)$ where $C$ is square.

By multiplying it (in the same spirit as in (2) by $C^{-1}$, one obtains $(I_k|E)=(I_k|C^{-1}D)$ ;

As a partial conclusion, matrices $B$ such that $B^TB=A$ correspond in a bijective way to k-dimensional subspaces in $\mathbb{R}^n$, thus can be parameterized in the same way as them, using $k \times (n-k)$ parameters.

Point 3) Consequently, $\frak{H}$ considered as a manifold (it is evidently not a vector space), has dimension $k(n-k)$. See for example Stack Exchange question (What is the dimension of this Grassmannian?).

Another reference linked to statistical applications: (http://www.cis.upenn.edu/~cis515/Turaga_Stiefel_2011.pdf).