# What is the relevance of this theorem? (Complete orthogonal decomposition)

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?

• Perhaps finding a triangular (rather than a diagonal) $A$ is computationally easier? Presumably such a decomposition is non-unique – Omnomnomnom Mar 1 at 14:34
• @Omnomnomnom: I see your point. But then perhaps that makes the algorithm that produces this decomposition interesting, not the theorem itself. – JLagana Mar 1 at 14:49
• I think that it is easy to immediately jump to SVD when you see $UAV=\begin{bmatrix} R & 0\\ 0 & 0 \end{bmatrix}$ since $U$ and $V$ are both unitary. However, SVD says there exists a factorization $A = URV^H$, where $U$ and $H$ are unitary, and $R$ is diagonal, so SVD makes the extra assumption that $R$ is diagonal, while the factorization you are asking makes the assumption that $R$ is triangular. I think it is just another matrix factorization that may or may not come in handy. – se2018 Mar 1 at 15:15