Eigensystems

Contentes

  1. Modules and classes

  2. Real Symmetric Matrices

  3. Complex Hermitian Matrices

  4. Real Nonsymmetric Matrices (>= GSL-1.9)

  5. Real Generalized Symmetric-Definite Eigensystems (>= GSL-1.10)

  6. Complex Generalized Hermitian-Definite Eigensystems (>= GSL-1.10)

  7. Real Generalized Nonsymmetric Eigensystems (>= GSL-1.10)

  8. Sorting Eigenvalues and Eigenvectors

Modules and classes

Real Symmetric Matrices

Workspace classes


Methods to solve eigensystems



  1. Singleton method of the GSL::Eigen module, GSL::Eigen::symm

    m = GSL::Matrix.alloc([1.0, 1/2.0, 1/3.0, 1/4.0], [1/2.0, 1/3.0, 1/4.0, 1/5.0],
                       [1/3.0, 1/4.0, 1/5.0, 1/6.0], [1/4.0, 1/5.0, 1/6.0, 1/7.0])
    eigval, eigvec = Eigen::symmv(m)
    
  2. Instance method of GSL::Matrix class

    eigval, eigvec = m.eigen_symmv
    

Complex Hermitian Matrices



Real Nonsymmetric Matrices (>= GSL-1.9)







Real Generalized Symmetric-Definite Eigensystems (GSL-1.10)

The real generalized symmetric-definite eigenvalue problem is to find eigenvalues lambda and eigenvectors x such that where A and B are symmetric matrices, and B is positive-definite. This problem reduces to the standard symmetric eigenvalue problem by applying the Cholesky decomposition to B: Therefore, the problem becomes C y = lambda y where C = L^{-1} A L^{-t} is symmetric, and y = L^t x. The standard symmetric eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvalues and eigenvectors of the generalized symmetric-definite eigenproblem are always real.




Complex Generalized Hermitian-Definite Eigensystems (>= GSL-1.10)

The complex generalized hermitian-definite eigenvalue problem is to find eigenvalues lambda and eigenvectors x such that where A and B are hermitian matrices, and B is positive-definite. Similarly to the real case, this can be reduced to C y = lambda y where C = L^{-1} A L^{-H} is hermitian, and y = L^H x. The standard hermitian eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvalues of the generalized hermitian-definite eigenproblem are always real.




Real Generalized Nonsymmetric Eigensystems (>= GSL-1.10)








Sorting Eigenvalues and Eigenvectors







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