I'm reading the book Matrix Analysis for Scientists and Engineers by Alan J. Laub. In the chapter about Canonical forms, the following theorem is presented:
Theorem 10.25 (Complete Orthogonal Decomposition): Let $A\in\mathbb{C}^{m\times n}_r$. Then there exist unitary matrices $U\in\mathbb{C}^{m\times m}$ and $V\in\mathbb{C}^{n\times n}$ such that $$UAV=\begin{bmatrix} R & 0\\ 0 & 0 \end{bmatrix} $$
where $R\in\mathbb{C}^{r\times r}_r$ is upper (or lower) triangular with positive diagonal elements.
Why is this relevant? It seems like I misunderstood this somehow, since to me this follows immediately from the SVD decomposition of $A$ (explained before in the book), and I expect $R$ to be diagonal, not only triangular. What am I missing?