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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

- Is this a well-known object?
- 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*? - 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.

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- Are “$n$ by $n$ matrices with rank $k$” an affine algebraic variety?
- Is this quotient ring $\mathbb{C}/\ker\phi$ integrally closed?
- Proving normality of affine schemes
- Is the hyperbola isomorphic to the circle?
- The importance of the structural morphism of a projective variety.

- Point reflection across a line
- Oh Times, $\otimes$ in linear algebra and tensors
- Winning strategies in multidimensional tic-tac-toe
- How to prove that homomorphism from field to ring is injective or zero?
- Divisor — line bundle correspondence in algebraic geometry
- How does one show a matrix is irreducible and reducible?
- For what value of h the set is linearly dependent?
- An eigenvector is a non-zero vector such that…
- What is the equation of the orthogonal group (as a variety/manifold)?
- Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?

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).

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