This animation shows a developing QR method in which one direct plane rotation is used for each active column. The rotation aligns that column with its leading coordinate direction, so every subdiagonal entry in the active part of the column becomes zero at once.
After each rotation, the completed row and column are set aside and the same construction is repeated on the trailing submatrix. The accumulated transposed rotations form the orthogonal factor Q, while the transformed matrix becomes upper triangular or upper trapezoidal R.
Under construction. The geometric construction is established, while numerical stability, sparse-matrix performance and possible hybrid algorithms remain open questions.
The animation follows the active-column process step by step. For each column, the method constructs the plane spanned by that column and the corresponding coordinate vector, rotates inside that plane, and leaves its orthogonal complement unchanged.
Unlike Givens rotations, which eliminate selected entries one at a time, this construction zeros the whole active part below the diagonal in a single rotation. Click the animation to open the full-size version.
At one active step, let the active column and the first active coordinate vector span a two-dimensional plane. A proper rotation in that plane aligns the column with the coordinate vector, while every vector perpendicular to the plane remains unchanged. The active column is therefore transformed into a scalar multiple of the coordinate vector.
The rotation can be written as U = I + B(J − I₂)BT, where the columns of B form an orthonormal basis for the rotation plane and J is the corresponding 2×2 rotation. Repeating the construction on successively deflated trailing submatrices produces R, while the product of the transposed rotations produces Q.
The method shares the one-column-at-a-time structure of Householder QR, but uses proper rotations rather than reflections. It also shares the rotational character of Givens QR, while replacing many coordinate-plane rotations by one direct rotation for each active column.
Related chapter: QRF, direct rotations
Related work: GraphMath Linear Algebra
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