《Linear Transformations in N-Dimensional Vector Space》

CHAPTER Ⅰ．LINEAR MANIFOLDS AND LINEAR TRANSFORMATIONS IN Bn1

1．Vectors and operations on vectors1

2．Linear manifolds8

3．Projections;orthonormal bases of a L．M．;complete orthonormal systems12

4．Products and sums of L．M．＇s17

5．Orthogonal complements;systems of homogeneous linear equations21

6．Linear transformations;systems of non-homogeneous linear equations24

7．Sums and products of L．T．＇s34

CHAPTER Ⅱ．SPECIAL LINEAR TRANSFORMATIONS IN Bn40

8．Hermitian transformations;eigen-values，eigen-manifolds and reduction40

9．Normal transformations and unitary transformations45

10．Projectors49

CHAPTER Ⅲ．SPECTRAL REPRESENTATION OF HERMITIAN TRANSFORMATIONS IN Bn58

11．Continuous functions in Bn58

12．Orthonormal systems of eigen-solutions of a H．T．The spectral representations60

13．Unitary transformations of a Hermitian form to principal axes67

14．Further maximal properties of the eigen-values．inequalities73

15．Functional calculus of H．T．＇s．The resolvent and its properties78

16．Transformations commutative with H．T．＇s．Normal transformations86

CHAPTER Ⅳ．SPECTRAL PROPERTIES OF LINEAR TRANSFORMATIONS IN Bn97

17．Eigen-values and principal manifolds97

18．The minimal polynomial103

19．Functional calculus for L．T．＇s．The resolvent108

20．Canonical bases of a principal manifold117

21．Jordan＇s canonical form for a nilpotent L．T．Commutativity122

22．Jordan＇s canonical form for any L．T．;elementary divisors and Segre characteristic;similar transformations130

23．Commutativity of L．T．＇s138

CHAPTER Ⅴ．VECTOR SPACES WITH POSITIVE HERMITIAN METRIC FORMS151

24．The spaces B＇n and Bn0151

25．Hermitian forms in B＇n and Bn0 and the pencil H－λG in Bn159

26．L．T．＇s that are Hermitian or normal in some B＇n164

27．Application to the dynamical theory of small oscillations166

Notes170

References184

Index of Authors190

General Index191