Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation and convention
- 1 Vector spaces
- 2 Linear mappings
- 3 Determinants
- 4 Scalar products
- 5 Real quadratic forms and self-adjoint mappings
- 6 Complex quadratic forms and self-adjoint mappings
- 7 Jordan decomposition
- 8 Selected topics
- 9 Excursion: Quantum mechanics in a nutshell
- Solutions to selected exercises
- Bibliographic notes
- References
- Index
7 - Jordan decomposition
Published online by Cambridge University Press: 18 December 2014
- Frontmatter
- Dedication
- Contents
- Preface
- Notation and convention
- 1 Vector spaces
- 2 Linear mappings
- 3 Determinants
- 4 Scalar products
- 5 Real quadratic forms and self-adjoint mappings
- 6 Complex quadratic forms and self-adjoint mappings
- 7 Jordan decomposition
- 8 Selected topics
- 9 Excursion: Quantum mechanics in a nutshell
- Solutions to selected exercises
- Bibliographic notes
- References
- Index
Summary
In this chapter we establish the celebrated Jordan decomposition theorem which allows us to reduce a linear mapping over ℂ into a canonical form in terms of its eigenspectrum. As a preparation we first recall some facts regarding factorization of polynomials. Then we show how to reduce a linear mapping over a set of its invariant subspaces determined by a prime factorization of the characteristic polynomial of the mapping. Next we reduce a linear mapping over its generalized eigenspaces. Finally we prove the Jordan decomposition theorem by understanding how a mapping behaves itself over each of its generalized eigenspaces.
Some useful facts about polynomials
Let P be the vector space of all polynomials with coefficients in a given field F and in the variable t. Various technical computations and concepts involving elements in P may be simplified considerably with the notion ‘ideal’ as we now describe.
Definition 7.1 A non-empty subset I ⊂ P is called an ideal of P if it satisfies the following two conditions.
f + g ∈ I for any f, g ∈ I.
fg ∈ I for any f ∈ P and g ∈ I.
Since F may naturally be viewed as a subset of P, we see that af ∈ I for any a ∈ F and f ∈ I. Hence an ideal is also a subspace.
Let g1, …, gk ∈ P. Construct the subset of P given by
{f1g1 + ··· + fkgk | f1, …, fk ∈ P}.
It is readily checked that the subset defined in (7.1.1) is an ideal of P. We may say that this deal is generated from g1, …, gk and use the notation I(g1, …, gk) to denote it.
- Type
- Chapter
- Information
- A Concise Text on Advanced Linear Algebra , pp. 205 - 225Publisher: Cambridge University PressPrint publication year: 2014