Given a basis of a finite dimensional vector space , we recall from Section 3.5 that the matrix of components of a linear operator with respect to this basis is defined by Eq. (3.6) as:
and under a transformation of basis,
where
the components of any linear operator transform by
The matrices and are inverse to each other, , and Eq. (4.4) can be written in matrix notation as a similarity transformation
The main task of this chapter will be to find a basis that provides a standard representation of any given linear operator, called the Jordan canonical form. This representation is uniquely determined by the operator and encapsulates all its essential properties. The proof given in Section 4.2 is rather technical and may be skipped on first reading. It would, however, be worthwhile to understand its appearance, summarized at the end ofthat section, as it has frequent applications in mathematical physics. Good references for linear operators and matrices in general are [1–3], while a detailed discussion of the Jordan canonical form can be found in [4].
It is important to realize that we are dealing with linear operators on free vector spaces. This concept will be defined rigorously in Chapter 6, but essentially it means that the vector spaces have no further structure imposed on them. A number of concepts such as ‘symmetric’, ‘hermitian’ and ‘unitary’, which often appear in matrix theory, have no place in free vector spaces. For example, the requirement that be a symmetric matrix would read in components, an awkward-looking relation that violates the rules given in Section 3.6. In Chapter 5 we will find a proper context for notions such as ‘symmetric transformations’ and ‘hermitian transformations’.
Contents
- 4.1 Eigenspaces and characteristic equations
- 4.2 Jordan canonical form
- 4.3 Linear ordinary differential equations
- 4.4 Introduction to group representation theory
Concept index
Terms by section. Links lead to the brown underlined definitions in the Chinese reading text.
4.1 Eigenspaces and characteristic equations
- invariant subspace不变子空间
- eigenvalue特征值
- eigenvector特征向量
- operator polynomial算符多项式
- characteristic equation特征方程
- characteristic polynomial特征多项式
- multiplicity重数
- minimal annihilating polynomial最小零化多项式
- elementary divisor初等因子
- trace迹
4.2 Jordan canonical form
4.3 Linear ordinary differential equations
- matrix exponential矩阵指数
- autonomous system自治系统
- critical point临界点
- stable node稳定结点
- unstable node不稳定结点
- saddle point鞍点
- stable focus稳定焦点
- unstable focus不稳定焦点
- vortex point涡点