Chapter 9: Matrix Decompositions and Beyond. In the chapter mind map this icon labels Rank-Revealing QR (RRQR) with Column Pivoting. The discussion below is excerpted and lightly edited from § Theorem: Rank-Revealing QR Decomposition in Mathematics for AI and Machine Learning.

For a matrix $A \in \mathbb{R}^{M \times N}$ with $\mathrm{rank}(A) = r$, there exists a permutation matrix $P \in \mathbb{R}^{M \times M}$ such that

where $P \in \mathbb{R}^{M \times M}$ is a permutation matrix (reordering rows), $Q \in \mathbb{R}^{M \times r}$ has orthonormal columns, $R_{11} \in \mathbb{R}^{r \times r}$ is upper triangular and well-conditioned, and $||R_{22}||$ is small (indicating that the last $N-r$ columns are nearly linearly dependent on the first $r$ columns).