Visualizing Matrix Decompositions in 2D and 3D

This page shows an engine for visualizing matrix decompositions and factorizations in 2D and 3D. The two examples below illustrate QR decomposition as a composition of an orthogonal transformation and an upper triangular transformation.

QR decomposition in 2D

In this example, the matrix A is written as A = QR. The upper triangular factor R shears and rescales the grid, while the orthogonal factor Q rotates it without changing lengths or angles.

2D geometric visualization of QR decomposition showing A equals Q times R with an upper triangular transformation followed by an orthogonal transformation

Image file: qr-2d-visualization.png

QR decomposition in 3D

The same structure appears in three dimensions. Here the factor R produces an upper triangular transformation, and the factor Q applies an orthogonal transformation, separating deformation from rigid motion.

3D geometric visualization of QR decomposition showing A equals Q times R with an upper triangular transformation followed by an orthogonal transformation

Image file: qr-3d-visualization.png

Related work: GraphMath Linear Algebra

Contact: info@graphmath.com