MATLAB Mathematics - SERC - Index of

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1 Matrices and Linear Algebra Eigenvalues An eigenvalue and eigenvector of a square matrix A are a scalar λ nonzero vector v that satisfy and a Av = λv This section explains: • Eigenvalue decomposition • Problems associated with defective (not diagonalizable) matrices • The use of Schur decomposition to avoid problems associated with eigenvalue decomposition Eigenvalue Decomposition With the eigenvalues on the diagonal of a diagonal matrix Λ and the corresponding eigenvectors forming the columns of a matrix V, you have AV = VΛ If V is nonsingular, this becomes the eigenvalue decomposition A = VΛV – 1 A good example is provided by the coefficient matrix of the ordinary differential equation in the previous section: A = 0 -6 -1 6 2 -16 -5 20 -10 The statement lambda = eig(A) produces a column vector containing the eigenvalues. For this matrix, the eigenvalues are complex: lambda = -3.0710 -2.4645+17.6008i -2.4645-17.6008i 1-38
Eigenvalues The real part of each of the eigenvalues is negative, so e λt approaches zero as t increases. The nonzero imaginary part of two of the eigenvalues, ± ω , contributes the oscillatory component, sin( ωt) , to the solution of the differential equation. With two output arguments, eig computes the eigenvectors and stores the eigenvalues in a diagonal matrix: [V,D] = eig(A) V = -0.8326 0.2003 - 0.1394i 0.2003 + 0.1394i -0.3553 -0.2110 - 0.6447i -0.2110 + 0.6447i -0.4248 -0.6930 -0.6930 D = -3.0710 0 0 0 -2.4645+17.6008i 0 0 0 -2.4645-17.6008i The first eigenvector is real and the other two vectors are complex conjugates of each other. All three vectors are normalized to have Euclidean length, norm(v,2), equal to one. The matrix V*D*inv(V), which can be written more succinctly as V*D/V, is within roundoff error of A. And, inv(V)*A*V, or V\A*V, is within roundoff error of D. Defective Matrices Some matrices do not have an eigenvector decomposition. These matrices are defective, or not diagonalizable. For example, A = [ 6 12 19 -9 -20 -33 4 9 15 ] For this matrix [V,D] = eig(A) 1-39
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3 Fast Fourier Transform (FFT) 2 x
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3 Fast Fourier Transform (FFT) 500
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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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5 Differential Equations Initial Va
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5 Differential Equations y0, yp0 Ve
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5 Differential Equations 1 0.8 0.6
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5 Differential Equations Example: A
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5 Differential Equations Function d
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5 Differential Equations u1(x,t) 1
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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 The vertices of o
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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 The following MAT
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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 Manipulating Spar
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Index comparing sparse and full mat
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Index H hb1dae demo 5-35 hb1ode dem
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Index nonstiff ODE examples rigid b
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Index LU factorization 6-30 minimum
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Index twobvp demo 5-63 two-dimensio
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