Spheres and orthogonal matrices as spaces of solutions to matrix equations

Question

For $a \in \mathbb{R}^n$, the solutions to

$$1=\sum_{i=1}^na_i^2$$

form an $(n-1)$-sphere in $\mathbb{R}^n$. Meanwhile, for $A \in \mbox{GL}_k (\mathbb{R})$, the solutions to

$$1=AA^T$$

are the real orthogonal matrices. For $A_i \in \mbox{GL}_k (\mathbb{R})$, what is the space of solutions to the following?

$$1=\sum_i^nA_iA_i^T$$

Is it related to a Stiefel manifold?

Answer 1

$$\mathrm I_k = \sum_{i=1}^n \mathrm X_i^\top \mathrm X_i = \begin{bmatrix} \mathrm X_1\\ \mathrm X_2\\ \vdots \\ \mathrm X_n\end{bmatrix}^\top \underbrace{\begin{bmatrix} \mathrm X_1\\ \mathrm X_2\\ \vdots \\ \mathrm X_n\end{bmatrix}}_{=: \tilde{\mathrm X}} = \tilde{\mathrm X}^\top \tilde{\mathrm X}$$

which defines the Stiefel manifold of $k$ orthonormal vectors in $\mathbb R^{nk}$.