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Lecture

Diagonalizability of 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 explores the concept of diagonalizability of matrices by analyzing the relationship between algebraic and geometric multiplicities of eigenvalues. The instructor explains how to determine if a matrix is diagonalizable by comparing these multiplicities and finding linearly independent eigenvectors. Through examples, the lecture demonstrates the process of finding eigenvectors and forming bases for eigenspaces. The importance of understanding the dimensions of eigenspaces and their implications for diagonalizability is emphasized. The lecture concludes by highlighting the key theorem that determines whether a matrix is diagonalizable based on the equality of algebraic and geometric multiplicities of eigenvalues.

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Instructors (2)

Nicolas Boumal

Jérôme Scherer

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Diagonalizable matrix

In linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix and a diagonal matrix such that , or equivalently . (Such , are not unique.) For a finite-dimensional vector space , a linear map is called diagonalizable if there exists an ordered basis of consisting of eigenvectors of .

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.

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.

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.

Companion matrix

In linear algebra, the Frobenius companion matrix of the monic polynomial is the square matrix defined as Some authors use the transpose of this matrix, , which is more convenient for some purposes such as linear recurrence relations (see below). is defined from the coefficients of , while the characteristic polynomial as well as the minimal polynomial of are equal to . In this sense, the matrix and the polynomial are "companions". Any matrix A with entries in a field F has characteristic polynomial , which in turn has companion matrix .

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