Eigenvalues and eigenvectors

Eigenvalues and eigenvectors
Important matrix properties
- Hermitian, orthogonality,…
Eigenvalue algorithm
- 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 ?)
- QR algorithm

Applications

collisional relaxation
- e.g. 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 population pyramid)
normal modes analysis (molecular vibrations)
PCA (principal component analysis)
Schrödinger equation in quantum mechanics, molecular orbitals (Hartree–Fock theory)

NumPy (more portability)
- numpy.linalg
  - linalg.eig(a) Compute the eigenvalues and right eigenvectors of a square array
SciPy (more complete wrapper on LAPACK fortran package)
- scipy.linalg
  - scipy.linalg.eig Solve an ordinary or generalized eigenvalue problem of a square matrix
  - Decompositions (LU, QR,…)

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
Eigenvectors and Eigenvalues: A deeper understanding Muhammad Abdullah Bilal, Analytics Vidhya, Medium, 04/03/2021