background information: I just started with the topic of singular values. Please be understanding.
groundwork:
So lets say we a matrix $ X^{(t)} $. $U_t$ is a basis for the columb spaces of $ X^{(t)} $. Hence theres a matrix $V_t$ such that $ X^{(t)} = U_tV_t^H $. We define the reduced gramian at $t$ as $ G_t = V_t^HV_t $.
From
$$X^{(t)}(X^{(t)})^H = U_tV_t^HV_tU_t^H = U_tG_tU_t^H$$
it follows that the singular values of $X^{(t)} $ are the square roots of the eigenvalues of the reduced Gramian $ G_t $, provided that $ U_t^HU_t = I $.
Is it possible to explain this in more detail?