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Lecture

Linear Algebra: Decomposition Theorems

In course

MATH-115(b): Advanced linear algebra II

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux du sujet. ...

Description

This lecture covers the proof of the primary decomposition theorem for endomorphisms of finite-dimensional vector spaces, showing the properties of invariant subspaces and nilpotent endomorphisms. It also explains the spectral polynomial and the concept of generalized eigenvectors.

Instructor

Marc Troyanov

Doctorat en mathématiques à l'Université de Genève (1987) - Postdoc à Paris et Salt-Lake City, puis professeur assistant à l'UQAM (Montréal). Venu en 1993 à l'EPFL comme professeur assistant, nommé MER en 1999 puis professeur titulaire en 2005. Domaine de recherche : géométrie différentielle, analyse sur les variétés et sur les espaces métriques.

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 A

Linear A is a writing system that was used by the Minoans of Crete from 1800 to 1450 BC to write the hypothesized Minoan language or languages. Linear A was the primary script used in palace and religious writings of the Minoan civilization. It was succeeded by Linear B, which was used by the Mycenaeans to write an early form of Greek. It was discovered by archaeologist Sir Arthur Evans. No texts in Linear A have yet been deciphered.

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.

Spectral theorem

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces.

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.

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