NLA Visualization
Explorable explanations for Numerical Linear Algebra. Geometry first.
Basics
Gram-Schmidt Algorithms
Classical Gram-Schmidt
Visualize orthogonalization using standard projections.
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Modified Gram-Schmidt
Visualize numerically stable orthogonalization.
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Householder Reflections
Visualize QR factorization using Householder reflections in 3D.
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Givens Rotations
Visualize QR factorization using Givens rotations in 3D.
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Conditioning Analyses
Conditioning of Matrix I
Understand how the pointwise condition number of A varies based on the input vector x.
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Conditioning of Matrix II
Visualize matrix perturbation geometry and deduced worst-case amplification.
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Conditioning of LSE I
Understand the geometry of the normal equations projection and how b is projected onto the plane defined by A.
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Conditioning of LSE II (Under Construction)
Sensitivity of the least squares solution to matrix perturbations.
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Iterative Methods
1. Power Iteration — Dominant Eigenvector
Visualizing how a vector converges to the dominant eigenvector.
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2. Power Iteration — Spectral Decay (rho(A)<1)
Visualizing how a vector shrinks toward the origin when the spectral radius is less than 1.
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3. Stationary Iterative Methods
Visualizing the iterative process and error decay in solving Amathbfx=mathbfb using matrix splitting.
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4. Arnoldi Iteration
Visualizing the expansion of the Krylov subspace block by block.
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5. Conjugate Gradient
Visualizing why CG finishes in n steps while GD zig-zags on an SPD matrix.
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6. Batch Rotations
Step-by-step explicit QR iteration forming R and RQ.
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7. Bulge Chasing
Step-by-step implicit QR iteration tracking the matrix bulge.
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8. Implicit Q Theorem
Visualizing why Bulge Chasing is equivalent to Explicit QR.
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9. Eigen Solver
Full end-to-end QR iteration algorithm for finding eigenvalues.
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