Householder Reflection — Geometric Interpretation

A concept note on using a reflection to align a vector with a coordinate direction in QR factorization

A Householder reflection is an orthogonal transformation that reflects space across a hyperplane. In QR factorization, it is used to send a chosen column vector to a coordinate direction, zeroing all entries below the pivot at once.

Unlike a Givens rotation, which eliminates one selected entry at a time, one Householder reflection acts on a whole vector direction. The transformation is global, but it still preserves lengths and angles because it is orthogonal.

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Visualization

The visual shows a 3×3 matrix A and its QR factorization. The original columns of A are shown with the corresponding transformed geometry after applying a Householder reflection.

Householder reflection visualization showing a 3 by 3 matrix, its QR factorization and the reflected column geometry.
Householder reflection aligns the active column with a coordinate direction while the other columns move under the same orthogonal transformation.

Concept

A Householder reflection has the form H = I − 2uuT, where u is a unit vector. The vector u is normal to the reflecting hyperplane.

Components lying in the reflecting hyperplane stay fixed, while the component in the u direction changes sign. Choosing u from the vector being reduced makes the reflection send that vector to a coordinate axis.

Role in QR factorization

In QR factorization, Householder reflections are applied successively to transform A into an upper triangular matrix R. The accumulated orthogonal transformations give the factor Q.

This gives a geometric interpretation of QR: the original column geometry is reflected step by step until it has triangular coordinate structure.

Key equations

H = I − 2uuᵀ, where ||u|| = 1
Hᵀ = H
HᵀH = I
H reflects across the hyperplane perpendicular to u
R = Hₖ ··· H₂ H₁ A
A = Q R

The symmetry equation says the transformation is a reflection. The orthogonality equation says it preserves lengths and angles.

Query phrases

  • Householder reflection geometric interpretation
  • Householder transformation QR factorization
  • reflection across a hyperplane in linear algebra
  • Householder QR visualization
  • orthogonal reflection matrix

References

Related concept: GraphMath — QR Decomposition, Geometric Interpretation in 2D and 3D

Related concept: GraphMath — Gram-Schmidt Orthogonalization, Stepwise Visualization in 2D and 3D

Related concept: GraphMath — Givens Rotation Algorithm, Geometric Interpretation in 3D