Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Eigenvalue Problems: Methods and Applications

In course

PHYS-332: Computational physics III

This course teaches the students practical skills needed for solving modern physics problems by means of computation. A number of examples illustrate the utility of numerical computations in various domains of physics. ...

Description

This lecture covers the basics of eigenvalue problems, including eigenvalue decomposition, normal and hermitian matrices, iterative methods such as the power method and inverse power method, Gershgorin circles theorem, Rayleigh quotient method, and numerical solutions for quantum particles in arbitrary potentials. It also discusses the application of iterative methods in computational physics, convergence properties, and examples of solving eigenvalue problems in connectivity networks and Google PageRank(TM) matrix.

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Algebra: Linear algebra

Related concepts (32)

Power iteration

In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix , the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, . The algorithm is also known as the Von Mises iteration. Power iteration is a very simple algorithm, but it may converge slowly.

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.

Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.

Inverse iteration

In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually similar to the power method. It appears to have originally been developed to compute resonance frequencies in the field of structural mechanics.

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Related lectures (47)

Eigenvalue Problems: Methods and Applications

Explores eigenvalue problems, iterative methods, and their applications in quantum physics and network analysis.

Eigenvalues and Optimization: Numerical Analysis Techniques

Discusses eigenvalues, their calculation methods, and their applications in optimization and numerical analysis.

Matrix Computations: Eigenvalues and Eigenvectors

Explores the complexity of matrix computations, focusing on eigenvalues and eigenvectors of symmetric matrices and the challenges in computing them.

Explicit Stabilised Methods: Applications to Bayesian Inverse Problems

Explores explicit stabilised Runge-Kutta methods and their application to Bayesian inverse problems, covering optimization, sampling, and numerical experiments.

Multigroup Theory: Main Equations and Numerical Solution