Analytic Fourier transform api

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[crowlogic]

restart;

Define the kernel and basis functions

K := (x, y) -> your_kernel_expression; # Replace your_kernel_expression with your kernel function K(x, y)
psi := n -> (x) -> basis_function_expression; # Replace basis_function_expression with your basis function psi_n(x)

Compute the matrix elements

MatrixK := Matrix(N, N, (m, n) -> int(int(psi(m)(x)*K(x, y)*psi(n)(y), y = 0 .. infinity), x = 0 .. infinity));

Solve the eigenvalue problem

Eigensystem := Eigenvalues(MatrixK, output = ['vectors']);

Output the results

Eigensystem;

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Alexander Farrugia

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Used matrices extensively in his PhD thesis.6y

If a matrix MM is diagonalizable, then M=PΛP−1M=PΛP−1, where ΛΛ is the diagonal matrix whose diagonal entries are the eigenvalues of MM and PP is the matrix whose columns are the eigenvectors of MM. We then have

M−1=(PΛP−1)−1=(P−1)−1Λ−1P−1=PΛ−1P−1M−1=(PΛP−1)−1=(P−1)−1Λ−1P−1=PΛ−1P−1

where Λ−1Λ−1 is the diagonal matrix whose diagonal entries are the reciprocals of the eigenvalues of MM. Note that, as a corollary, M−1M−1 exists if and only if none of the eigenvalues of MM are zero.

Another relation between matrix inversion and eigenvalues is the following. Let MM be any square matrix and p(x)=anxn+an−1xn−1+⋯+a1x+a0p(x)=anxn+an−1xn−1+⋯+a1x+a0 be the characteristic polynomial of MM (whose roots are the eigenvalues of MM). Then, by the Cayley-Hamilton theorem, we have

anMn+an−1Mn−1+⋯+a1M+a0I=0.anMn+an−1Mn−1+⋯+a1M+a0I=0.

If M−1M−1 exists, then

anMn−1+an−1Mn−2+⋯+a1I+a0M−1=0anMn−1+an−1Mn−2+⋯+a1I+a0M−1=0

so

M−1=−1a0(anMn−1+an−1Mn−2+⋯+a1I).M−1=−1a0(anMn−1+an−1Mn−2+⋯+a1I).

Note that M−1M−1 exists if and only if a0≠0a0≠0. Since a0=(−1)ndet(M)a0=(−1)ndet(M), this is equivalent to stating the well-known result that M−1M−1 exists if and only if det(M)≠0det(M)≠0.

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