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1 Matrices and Linear Algebra Singular Value Decomposition A singular value and corresponding singular vectors of a rectangular matrix A are a scalar σ and a pair of vectors u and v that satisfy Av A T u = = σu σv With the singular values on the diagonal of a diagonal matrix Σ and the corresponding singular vectors forming the columns of two orthogonal matrices U and V, you have AV A T U = = UΣ V Σ Since U and V are orthogonal, this becomes the singular value decomposition A = U Σ V T The full singular value decomposition of an m-by-n matrix involves an m-by-m U, an m-by-n Σ , and an n-by-n V. In other words, U and V are both square and Σ is the same size as A. If A has many more rows than columns, the resulting U can be quite large, but most of its columns are multiplied by zeros in Σ . In this situation, the economy sized decomposition saves both time and storage by producing an m-by-n U, an n-by-n Σ and the same V. The eigenvalue decomposition is the appropriate tool for analyzing a matrix when it represents a mapping from a vector space into itself, as it does for an ordinary differential equation. On the other hand, the singular value decomposition is the appropriate tool for analyzing a mapping from one vector space into another vector space, possibly with a different dimension. Most systems of simultaneous linear equations fall into this second category. If A is square, symmetric, and positive definite, then its eigenvalue and singular value decompositions are the same. But, as A departs from symmetry and positive definiteness, the difference between the two decompositions increases. In particular, the singular value decomposition of a real matrix is always real, but the eigenvalue decomposition of a real, nonsymmetric matrix might be complex. 1-42
Singular Value Decomposition For the example matrix A = 9 4 6 8 2 7 the full singular value decomposition is [U,S,V] = svd(A) U = -0.6105 0.7174 0.3355 -0.6646 -0.2336 -0.7098 -0.4308 -0.6563 0.6194 S = 14.9359 0 0 5.1883 0 0 V = -0.6925 0.7214 -0.7214 -0.6925 You can verify that U*S*V' is equal to A to within roundoff error. For this small problem, the economy size decomposition is only slightly smaller: [U,S,V] = svd(A,0) U = -0.6105 0.7174 -0.6646 -0.2336 -0.4308 -0.6563 S = 14.9359 0 0 5.1883 V = -0.6925 0.7214 -0.7214 -0.6925 Again, U*S*V' is equal to A to within roundoff error. 1-43
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3 Fast Fourier Transform (FFT) Func
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4 Function Functions Function Summa
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4 Function Functions You can also c
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4 Function Functions 25 20 15 10 5
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4 Function Functions Minimizing Fun
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4 Function Functions To try fminsea
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4 Function Functions Plotting the R
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4 Function Functions The number of
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4 Function Functions and displays t
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4 Function Functions end function s
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4 Function Functions 100 80 60 40 2
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4 Function Functions You can verify
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4 Function Functions Troubleshootin
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4 Function Functions A three-dimens
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4 Function Functions Parameterizing
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5 Differential Equations Initial Va
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5 Differential Equations y0, yp0 Ve
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5 Differential Equations For exampl
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5 Differential Equations 1 0.8 0.6
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5 Differential Equations 2.5 Soluti
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5 Differential Equations 2 u′ i =
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5 Differential Equations Example: A
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5 Differential Equations Summary of
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5 Differential Equations Problem Si
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5 Differential Equations Function d
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6 Sparse Matrices Function Summary
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6 Sparse Matrices Function Summary
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6 Sparse Matrices This matrix requi
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6 Sparse Matrices S = (3,1) 1 (2,2)
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6 Sparse Matrices Now F = full(S) d
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6 Sparse Matrices Importing Sparse
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6 Sparse Matrices west0479 west0479
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6 Sparse Matrices The find Function
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6 Sparse Matrices of the rows and c
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6 Sparse Matrices 0 10 20 30 40 50
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6 Sparse Matrices 0 500 1000 1500 2
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6 Sparse Matrices simply sparse(m,n
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6 Sparse Matrices Similarly, S(:,p)
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6 Sparse Matrices 0 Original 0 Reve
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6 Sparse Matrices QR Factorization
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6 Sparse Matrices Selected Bibliogr
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Index LU factorization 6-30 minimum
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