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Algebraic Eigenvalue Problems
Published in Karan S. Surana, Numerical Methods and Methods of Approximation, 2018
In this method we construct the characteristic polynomial p(λ) corresponding to the eigenvalue problem (either SEVP or GEVP). The roots of the characteristic polynomial are the eigenvalues. For each eigenvalue we determine the eigenvector using the eigenvalue problem.
Linear Algebra Problems
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
The definition of the roots of the characteristic polynomial discussed earlier is exactly the same as the eigenvalues. If the characteristic polynomial can be exactly known, the function can also be used in evaluating the eigenvalues of the matrix.
Mathematical Background
Published in Ferenc Szidarovszky, A. Terry Bahill, Linear Systems Theory, 2018
Ferenc Szidarovszky, A. Terry Bahill
[The Cayley-Hamilton Theorem] LetAbe an n × n real or complex matrix, and let φ denote its characteristic polynomial. Thenφ(A) = O,
Leading students towards the formal world of mathematical thinking: a mathematician’s reflections on teaching eigentheory
Published in International Journal of Mathematical Education in Science and Technology, 2019
Sepideh Stewart, Jonathan Epstein, Jonathan Troup
The instructor made the decision to use Day 3 not for the next IOLA task, but instead to synthesize the various embodied, symbolic and formal aspects of eigentheory that the students have so far encountered. To do so, he used exclusively a lecture teaching style. First, he showed how the black and blue coordinate matrix representations of the transformation from those tasks are related by conjugation by the change of coordinate matrix. Next, starting with the standard coordinate representation of the linear transformation, he used GeoGebra to demonstrate visually the effect to the linear transformation on vectors in the unit circle, and in particular how it exactly stretches some, but not all, directions. At this point, he reiterated the eigenvalue and eigenvector definitions, and derived the standard way of computing them from the characteristic polynomial and finding the nullspace of A – λI. From here, he presented a series of examples including the transformation from the IOLA tasks, an eigenspace with more than one dimension, and the differentiation operator acting on function spaces.
Dynamic modeling of air traffic emissions with a two variable system
Published in International Journal of Sustainable Transportation, 2021
Francisco A. Buendia-Hernandez, Francisco J. Alvarez-Garcia, Maria J. OrtizBevia, Antonio RuizdeElvira
In the case of the linear ODE system, the coefficients of (1) were collected into a dyna-mical matrix The characteristics of the corresponding dynamical systems can be obtained from a stability analysis determined from the root of the characteristic polynomial, obtained from the dynamical matrix A of the system (Perko, 2001), as detailed in the Appendix 1.