QR decomposition in 2D
In two dimensions, R first applies an upper triangular transformation to the grid, and Q then applies an orthogonal transformation. This separates deformation from rigid motion.
QR Decomposition — Geometric Interpretation in 2D and 3D
QR decomposition writes a matrix as A = Q R, where R is upper triangular and Q is orthogonal. Geometrically, this separates two different effects: R performs a shear-and-scale type transformation in a fixed coordinate frame, while Q applies a rigid rotation or reflection that preserves lengths and angles.
QR decomposition in 3D
The same structure appears in three dimensions. The upper triangular factor R changes shape in a coordinate-dependent way, and the orthogonal factor Q then rotates or reflects the result without changing lengths and angles.
Concept
For a square matrix with independent columns, QR decomposition factors the matrix into A = Q R, where the columns of Q form an orthonormal basis and R is upper triangular.
This means the action of A can be understood as a composition: first apply R, then apply Q. Because matrix products act right to left, A = Q R means that R happens first and Q happens second.
Structure
The factor R is upper triangular, so it encodes a hierarchical deformation in the chosen coordinate system. The factor Q is orthogonal, so it preserves inner products, lengths and angles.
This decomposition separates shape-changing behavior from rigid motion explicitly.
References
2D reference: Wikipedia — QR decomposition, square matrix section
3D reference: Wikipedia — QR decomposition, example section
Related visual: Gram-Schmidt Orthogonalization — Stepwise Visualization in 2D and 3D
Related chapter: QRF, Gram-Schmidt, part 1
Related work: GraphMath Linear Algebra
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