Lecture

Eigenvalues and Stability

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In course

MATH-251(a): Numerical analysis

This course presents numerical methods to solve mathematical problems such as systems of linear and nonlinear equations, function approximation, integration and differentiation, and differential equations. ...

Description

This lecture covers the concept of eigenvalues and stability in linear systems, discussing the conditions for conditional and unconditional stability, as well as the school case scenario. It also explores the use of eigenvectors in progressive and retrograde systems, providing error estimation techniques and error checks.

Instructors (4)

Fabio Nobile

Teaching & PhD PhD Students Veronica Montanaro, Matteo Raviola, Francesca Bettinelli, Eliott Van Dieren, Riccardo Saporiti, Giacomo Mossinelli, Noé Stauffer, Fabio Zoccolan Past EPFL PhD Students Diane Sylvie Guignard, Francesco Tesei, Eleonora Musharbash, Matthieu Claude Martin, Davide Pradovera, Sundar Subramaniam Ganesh, Eva Vidlicková, Juan Pablo Madrigal Cianci, Andrea Zanoni, Thomas Simon Spencer Trigo Trindade Past EPFL PhD Students as codirector Michele Pisaroni Courses Advanced analysis II - ve

Daniel Kressner

Pablo Antolin Sanchez

Teaching & PhD PhD Students Philipp Christoph Weder, Davor Dobrota Past EPFL PhD Students as codirector Ondine Chanon Courses Analysis III MATH-202(c) The course studies the fundamental concepts of vector analysis and Fourier-Laplace analysis with a view to their use in solving multidisciplinary problems in scientific engineering. Analysis III (for IC) MATH-203(d) The course studies the fundamental concepts of vector analysis and Fourier-Laplace analysis with a view to thei

Espen Sande

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

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.

Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.

Eigendecomposition of a matrix

In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.

Generalized eigenvector

In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis. There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.

Eigenvalue algorithm

In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Eigenvalues and eigenvectors and Generalized eigenvector Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.

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