3blue1brownGeometric routes to QR factorization
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Tags: linear-algebravisualizationmatricesqr-factorizationmatrix-algorithms
Visual comparison of four geometric constructions for QR factorization: Gram–Schmidt orthogonalization, Givens rotations, Householder reflections, and Direct Rotations.
The first three are standard QR constructions. Direct Rotations is a much less commonly presented construction that we arrived at independently while developing these visualizations; we later found an earlier related formulation in the literature.
The same full-rank matrix and the same flat rank-deficient matrix are used for all four constructions. The animations show how Gram–Schmidt reshapes the columns of A into Q, while Givens rotations, Householder reflections, and Direct Rotations transform A into the upper triangular matrix R.
In 2D, Direct Rotation coincides with the single Givens rotation. In higher dimensions, Givens uses successive coordinate-plane rotations, while Direct Rotations uses one plane tailored to each active column and its target axis. Direct Rotations is also closely related to Householder QR, despite using proper rotations rather than reflections.
The entry is intended for students learning linear algebra, mostly at the undergraduate level, as well as instructors of those courses.
Created by Eteri Byazrova and Yuri Morozov for GraphMath / Linear Algebra World.
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This seems very interesting. High clarity, and the introduction does a good job of explaining the motivation. I don’t actually know how novel this is (tell me in the piece!), and there isn’t much of a through-line to help me understand where and when to use these tools. I think a dose of narrative by example would have gone a long way here.
Thank you for such great animations, these are great visual complements when teaching QR decomposition. The illustration is also clear and complete.
Just some bullet points. No motivation, no narrative, no explanations, no context.
I enjoyed the videos, but I found the notation wasn’t clearly defined and in general the exposition was lacking, making it harder to follow the post. In terms of the target audience, I also thought it might make sense to split by topics instead of by the current sections, so that depending on where the student is in learning linear algebra, they could look at just that section.