Introduction - UCONN

THE MINIMAL POLYNOMIAL AND SOME APPLICATIONSKEITH easiest matrices to compute with are the diagonal ones. The sum and product ofdiagonal matrices can be computed componentwise along the main diagonal, and takingpowers of a diagonal matrix is simple too. All the complications of matrix operations aregone when working only with diagonal matrices. If a matrixAis not diagonal but can beconjugated to a diagonal matrix, sayD:=PAP 1is diagonal, thenA=P 1 DPsoAk=P 1 DkPfor all integersk, which reduces us to computations with a diagonal matrix. Inmany applications of linear algebra ( , dynamical systems, differential equations, Markovchains, recursive sequences) powers of a matrix are crucial to understanding the situation,so the relevance of knowing when we can conjugate a nondiagonal matrix into a diagonalmatrix is want look at the coordinate-free formulation of the idea of a diagonal matrix, whichwill be called a diagonalizable operator. There is a special polynomial, the minimal polyno-mial (generally not equal to the characteristic polynomial), which will tell us exactly whena linear operator is diagonalizable.

THE MINIMAL POLYNOMIAL AND SOME APPLICATIONS KEITH CONRAD 1. Introduction The easiest matrices to compute with …

Fullscreen Download

Tags:

Introduction

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Spam in document Broken preview Other abuse

Transcription of Introduction - UCONN

Documents from same domain

Introduction The Divisibility Relation

www.math.uconn.edu

DIVISIBILITY AND GREATEST COMMON DIVISORS KEITH CONRAD 1. Introduction We will begin with a review of divisibility among integers, mostly to set some notation

Introduction, Relations, Introduction the divisibility relation, Divisibility

GALOIS THEORY AT WORK: CONCRETE EXAMPLES

www.math.uconn.edu

galois theory at work: concrete examples 5 (4) A nontrivial nite p-group has a subgroup of index p. The rst property is a consequence of the intermediate value theorem.

Theory, Galois theory, Galois

The Trace and Norm for a Galois extension - UCONN

www.math.uconn.edu

The Trace and Norm for a Galois extension Let L=Kbe a nite Galois extension, with Galois group G= Gal(L=K). We can express characteristic polynomials, traces, and norms for the extension L=Kin terms of G. Theorem 1.1. When L=Kis a nite Galois extension with Galois group Gand 2L, ... From Galois theory,

Theory, Extension, Galois theory, Galois, Galois extension

Introduction -automorphism 2 Example 1.1. f - UCONN

www.math.uconn.edu

(1.3) FˆLAut(L=F); HˆAut(L=LH): That is, F is contained in the set of elements xed by the automorphisms of Lwhich x Fand His contained in the set of automorphisms of Lwhich x the elements xed by H. (This is practically a tautology, but you may need to read through that two times to see it.)

Introduction, Example, Introduction automorphism 2 example, Automorphism