Every matrix taken apart: turn, stretch, turn. Singular values, the most important pieces of a matrix, image compression, principal component analysis and the pseudoinverse. 7 short slides, then worked examples and problems. Each slide has an everyday picture, a worked example and Python you can run right on the page.
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Lesson 8Turn, stretch, turnThe SVD: every matrix is a turn, a stretch and a turnOpen slide →01 · Strang §7.4 · The geometry of the SVDThe SVD says every matrix turns, stretches along perpendicular axes, then turns again: A = UΣVᵀ. The unit circle becomes an ellipse. Strang §7.4.
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Lesson 8Singular valuesFinding singular values from AᵀAOpen slide →02 · Strang §7.2 · Bases and matrices in the SVDSingular values are square roots of the eigenvalues of AᵀA; its eigenvectors are the v's, and u = Av/σ. Worked examples and Python. Strang §7.2.
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Lesson 8Rank-one piecesSplitting a matrix into rank-one piecesOpen slide →03 · Strang §7.2 · The pieces of the SVDThe SVD writes any matrix as a sum of simple rank-one pieces σuvᵀ, ordered from most to least important. Worked examples and Python. Strang §7.2.
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Lesson 8Image compressionCompressing an image with the SVDOpen slide →04 · Strang §7.1 · Image processing by linear algebraA grey image is a matrix. Keep only its largest singular values and the picture survives with far fewer numbers. Interactive Python example. Strang §7.1.
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Lesson 8PCAPCA: finding the main direction of dataOpen slide →05 · Strang §7.3 · Principal component analysisPCA finds the direction where data spreads most: center the data, take the first direction from the SVD. Interactive example with noisy robot positions. Strang §7.3.
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Lesson 8PseudoinverseThe pseudoinverse and the condition numberOpen slide →06 · Strang §7.4 · The pseudoinverseThe pseudoinverse A⁺ = VΣ⁺Uᵀ gives the best least-squares answer for any matrix, and σ₁/σₙ measures sensitivity. Python examples. Strang §7.4.
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Lesson 8RecapWhat you can now do.Open slide →07 · Recap · ReadingChapter 7 on one page: A = UΣVᵀ as turn-stretch-turn, singular values from AᵀA, rank-one pieces, image compression, PCA and the pseudoinverse.
