Transcription of Chapter 7 TheSingularValueDecomposition(SVD)
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Chapter 7 TheSingularValueDecomposition(SVD)
Chapter 7 The Singular Value Decomposition (SVD) 1 The SVD produces orthonormal bases of v s and u s for the four fundamental Using those bases, A becomes a diagonal matrix and Avi= iui: i= singular The two-bases diagonalization A = U VToften has more information than A = X X U VTseparates A into rank-1 matrices 1u1vT1+ + rurvTr. 1u1vT1is the largest! Bases and Matrices in the SVDThe Singular Value Decomposition is a highlight of linear algebra. A is any m by n matrix,square or rectangular. Its rank is r. We will diagonalize this A, but not by X eigenvectors in X have three big problems: They are usually not orthogonal, thereare not always enough eigenvectors, and Ax = x requires A to be a square matrix.
7.1. Bases and Matrices in the SVD 383 Example 2 If A = xyT (rank 1) with unit vectorsx and y, what is the SVD of A? Solution The reduced SVD in (2) is exactly xyT, with rank r = 1.It has u1 = x and v1 = y andσ1 = 1. For the full SVD, complete u1 = x to an orthonormal basis of u’ s, and complete v1 = y to an orthonormalbasis of v’s. No newσ’s, onlyσ1 = 1.
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