QR Factorization by Direct Plane Rotations

A concept note on a developing QR method that uses one proper rotation for each active column

This QR construction uses one direct plane rotation for each active column. At every step, the active column is rotated into its leading coordinate direction, so the whole active part below the diagonal becomes zero at once.

The method combines features of two familiar QR approaches. Like Householder QR, it works one column at a time. Like Givens QR, it uses rotations rather than reflections. The main idea is to replace many coordinate-plane rotations by one carefully chosen rotation in the plane spanned by the active column and its target coordinate vector.

Under construction. The geometric construction is clear, while questions about numerical stability, sparse-matrix performance and possible hybrid algorithms remain open.

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Visualization

The animation shows the method on a matrix with several columns. For each active column, the algorithm identifies the rotation plane, performs one proper rotation, and then continues on the deflated trailing submatrix.

Animated geometric visualization of QR factorization by direct plane rotations, showing one rotation aligning each active column with a coordinate direction.
One direct plane rotation aligns the active column with its coordinate direction, making all subdiagonal entries in that active column equal to zero at once. Click the image to open the full-size version.

Concept

Suppose the current active column is a and its target direction is the first active coordinate vector e₁. These two vectors span a two-dimensional plane. Inside that plane, there is a proper rotation that sends a to a positive multiple of e₁. Every vector perpendicular to the plane stays unchanged.

Because the whole active column is rotated into the coordinate direction in one step, all active entries below the pivot position disappear simultaneously. After that, the completed row and column are deflated, and the same construction is repeated on the trailing submatrix.

How it differs from Givens and Householder QR

A Givens method uses many coordinate-plane rotations and typically removes one selected entry at a time. A Householder method uses one reflection per active column. This direct-rotation method instead uses one proper rotation per active column.

So it keeps the rotational character of Givens QR but changes the scope of each step. At the same time, it keeps the one-column-at-a-time structure of Householder QR but replaces reflections by rotations.

Key equations

At one active step: U a = ||a|| e₁
U = I + B(J − I₂)Bᵀ
J = [ [cos θ, −sin θ], [sin θ, cos θ] ]
Uᵀ U = I, det(U) = 1
R = Uₖ ··· U₂ U₁ A
Q = U₁ᵀ U₂ᵀ ··· Uₖᵀ
A = Q R

The matrix B contains an orthonormal basis for the rotation plane, while J is the corresponding 2×2 rotation inside that plane. Repeating the step on successively smaller trailing submatrices produces the full factorization.

Query phrases

  • QR factorization by direct plane rotations
  • rotation-based QR decomposition
  • direct rotation QR method
  • QR one rotation per active column
  • geometric interpretation of QR by rotations