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Lecture

Decomposition of Linear Operators

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 decomposition of linear operators, including the primary decomposition theorem, properties of eigenspaces, and diagonalizability. It also discusses the minimal polynomial, direct sum decomposition, and the Cayley-Hamilton theorem.

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)

Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers.

Polynomial

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.

Direct sum

The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise.

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 .

Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the notion. The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers).

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