====== Eigenvalues and eigenvectors ======

  * [[wp>Eigenvalues_and_eigenvectors|Eigenvalues and eigenvectors]]
  * Important matrix properties
    * Hermitian, orthogonality,...
  * [[wp>Eigenvalue_algorithm|Eigenvalue algorithm]]
    * [[wp>Power_iteration|Power iteration]], a simple numerical algorithm producing a number $\lambda$, the greatest (in absolute value) eigenvalue of a matrix $A$, and the corresponding eigenvector $v$, such that $Av=\lambda v$.
    * LR algorithm, developed by Heinz Rutishauser (1958 ?)
    * [[wp>QR_algorithm|QR algorithm]]

===== Applications =====
  * collisional relaxation
    * //e.g.// [[https://pubs.acs.org/doi/abs/10.1021/ed066p142|Collisional relaxation via eigenfunction-eigenvalue expansion: Analysis of a simple case]] Wendell Forst, J. Chem. Educ., 1989, 66 (2), p 142 DOI: 10.1021/ed066p142
  * population dynamics, evolution (stationary [[wp>population pyramid]])
    * [[wp>Leslie_matrix|Leslie matrix]]
    * [[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.133.7164&rep=rep1&type=pdf|Leslie Matrix Model in Population Dynamics - CiteSeerX]]
    * [[http://personal.denison.edu/~ludwigl/2011populationdynamicsandma.pdf]],...
  * normal modes analysis (molecular vibrations)
  * PCA (principal component analysis)
  * Schrödinger equation in quantum mechanics, molecular orbitals (Hartree–Fock theory)

===== Python libraries =====
  * NumPy (more portability)
    * [[https://docs.scipy.org/doc/numpy/reference/routines.linalg.html|numpy.linalg]]
      * [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eig.html|linalg.eig(a)]] Compute the eigenvalues and right eigenvectors of a square array
  * SciPy (more complete wrapper on [[wp>LAPACK|LAPACK]] fortran package)
    * [[https://docs.scipy.org/doc/scipy/reference/linalg.html|scipy.linalg]]
      * [[https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eig.html|scipy.linalg.eig]] Solve an ordinary or generalized eigenvalue problem of a square matrix
      * [[https://docs.scipy.org/doc/scipy/reference/linalg.html#decompositions|Decompositions]] (LU, QR,...)


===== References =====

  * Numerical recipes, The Art of Scientific Computing 3rd Edition, William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, 2007, isbn: 9780521880688
    * [[http://numerical.recipes/]]
    * [[http://www2.units.it/ipl/students_area/imm2/files/Numerical_Recipes.pdf]], chapter 11 Eigensystems p 456...
    * [[http://apps.nrbook.com/empanel/index.html#]]
  * Python NumPy vs SciPy : //cf.// [[https://stackoverflow.com/questions/6684238/whats-the-fastest-way-to-find-eigenvalues-vectors-in-python]]
  * [[https://medium.com/analytics-vidhya/eigenvectors-and-eigenvalues-a-deeper-understanding-c715f8ded4c7|Eigenvectors and Eigenvalues: A deeper understanding]] Muhammad Abdullah Bilal, Analytics Vidhya, Medium, 04/03/2021