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

Lecture

Eigenvalues and Eigenvectors: Polynomials and Matrices

In course

MATH-111(e): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications. ...

Description

This lecture covers the concept of eigenvalues and eigenvectors, focusing on the characteristic polynomial of a matrix. The instructor explains how to determine eigenvalues and eigenvectors, illustrating with examples of 2x2 and 3x3 matrices. The lecture delves into the algebraic and geometric multiplicities of eigenvalues, emphasizing the importance of the characteristic polynomial in determining the number of eigenvalues. The discussion extends to the limitations of using matrix operations like summation and scalar multiplication to calculate eigenvalues. The instructor also highlights the significance of understanding the algebraic and geometric multiplicities of eigenvalues in solving matrix problems.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Nicolas Boumal

Jérôme Scherer

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (26)

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.

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.

Matrix multiplication

In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.

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.

Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

Related lectures (42)

Algebraic Multiplicity, Geometric Multiplicity MOOC: Linear Algebra (Part 2)

Explores algebraic and geometric multiplicities of eigenvalues in linear algebra.

Diagonalization of Linear Maps

Explores the diagonalization of linear maps by finding a basis formed by eigenvectors.

Diagonalization of Linear Transformations

Explains the diagonalization of linear transformations using eigenvectors and eigenvalues to form a diagonal matrix.

Eigenvalues and Eigenvectors

Covers eigenvalues, eigenvectors, characteristic polynomial, and geometric multiplicities in linear transformations.

Eigenvalues and Eigenvectors in 3D