Partial isometry, Isometry and Gram matrix

Question

Suppose I have a set of matrices $\mathcal{H}$ s.t. $\forall H\in\mathcal{H}, \ H = CU$ where $C$ is a fixed matrix and $U$ is an orthonormal matrix. We know that this set can be characterized completely by the Gram matrix, i.e. given any set of matrices ${H_1, \ldots, H_n}$ we can check if $H_i \in \mathcal{H}$ by calculating $G_i = H_i H^{\dagger}_i$ and check whether $G_i = C^2$. Now, suppose that given $H_1, H_2$ we want to check whether there exists a partial isometry such that $H_1 U_1 = H_2 U_2$ where $U_i = W_i S_r V^{\dagger}_i$ with $W_i, V_i$ orthonormal and $S_r = diag(I_{r\times r},0_{n-r \times n-r})$. Assuming that $r$ is known, is there any simple way to verify such a relation?

Answer 1

$H_1 = CW_1$ and $H_2 = CW_2$,$\implies H_1W_1^TW_2 = H_2 $. So $W_1^TW_2$ is your partial isometry. So any two matrix can be related in the way you mentioned ?

Good luck !