URV decomposition

For any matrix, $$\mathbf A$$ with rank $$r$$, there are orthogonal matrices $$\mathbf U$$ and $$\mathbf V$$ and a nonsingular matrix $$\mathbf C$$ such that (see p407 )

$$\mathbf{A=URV}^T = \mathbf U \begin{pmatrix}\mathbf C & \mathbf 0 \\ \mathbf 0 & \mathbf 0\end{pmatrix} \mathbf V^T,$$ such that
 * the first $$r$$ columns of $$\mathbf U$$ are an orthonormal basis for $$R(\mathbf A)$$
 * remaining columns of $$\mathbf U$$ are an orthonormal basis for $$N(\mathbf A^T)$$
 * the first $$r$$ columns in $$\mathbf V$$ are an orthonormal basis for $$R(\mathbf A^T)$$
 * remaining columns of $$\mathbf V$$ are an orthonormal basis for $$N(\mathbf A)$$

What is the point of URV factorization?
This is an extremely powerful theorem, particularly because it is not only a theoretical result -- it can be easily operationalized through the QR decomposition (see here). In fact even the theoretical implication of this decomposition are quite significant, e.g., see this post.

Moreover, URV directly leads to the development of SVD. In fact, the latter is nothing but a special case where the matrix $$\mathbf R$$ is diagonal (and the matrices $$\mathbf U$$ and $$\mathbf V$$ are unique.