Column-Pivoted QR Rank Decisions

Column pivoting is a modification to the QR decomposition that rearranges the columns of the input matrix to process the most dominant directions first. At each step, it selects the column with the largest residual norm. This technique is highly effective for discovering the numerical rank of a matrix, as the diagonal entries of the resulting upper triangular matrix R reveal the significance of each successive basis vector.

Experiment

You can interact with the column-pivoted QR factorization using the interactive module.

1. Set Small scale to "0.1".
2. Set Rank threshold to "0.05".
3. Observe that the Estimated numerical rank is 2, while the Exact rank is 3, because the third diagonal magnitude drops below the relative threshold.
4. Change Small scale to "1". The columns are now well-separated in magnitude, and the third diagonal is 0.7071067811865476, which is larger than 0.05 * 2.0. The estimated rank is now 3.
5. Change Small scale to "0" (with rank threshold "0.05"). The exact rank drops to 2, and the third diagonal magnitude becomes exactly 0.0. The numerical rank correctly estimates 2.