Row norms are invariant across factorizations $UU^\top=ZZ^\top$ (Claim C.2)

For natural numbers $d$ and $r$, given two real $d \times r$ matrices $U$ and $Z$ (with rows indexed by $\{1, \dots, d\}$ and columns by $\{1, \dots, r\}$) satisfying the hypothesis $U U^\top = Z Z^\top$, and given a row index $i \in \{1, \dots, d\}$, the theorem asserts that the Euclidean norm of the $i$-th row of $U$ equals the Euclidean norm of the $i$-th row of $Z$:

where the row norm is the bundle's rowNorm, defined as the square root of the sum of the squares of the row's entries (the square root being the real square root, which returns $0$ on negative inputs — irrelevant here since the sum of squares is nonnegative). The matrices $U$ and $Z$ are explicit universally quantified arguments, the dimensions $d$ and $r$ are implicit universally quantified natural numbers, and the conclusion is stated for the one given index $i$ (equivalently, since $i$ is universally quantified, for every row index). No rank, incoherence, or nonzero-dimension assumptions are made; the statement includes the degenerate cases $d = 0$ (no row indices exist, so the statement is vacuous) and $r = 0$ (both row norms are $\sqrt{0} = 0$, and the hypothesis $U U^\top = Z Z^\top$ holds automatically since both sides are the zero $d \times d$ matrix). Note the hypothesis is an exact matrix equality, not an approximation, and the conclusion is an exact equality of real numbers.