Contents
Chapter1ReviewandMiscellanea:BasicConceptsinLinearAlgebra1
11MatrixConceptandSpecialMatrices1
12MatrixAlgebra3
13EigenvalueandEigenvector7
14Reading:CayleyandSylvester10
15SelfAssessmentExercises11
Chapter2LinearSpaceandInner-productSpace13
21LinearSpace(VectorSpace)13
22BasisandDimensionofVectorSpace17
23Subspaces20
24InnerProductSpace21
25Reading:GramandSchmidt24
26SelfAssessmentExercises25
Chapter3LinearTransformation27
31Introduction27
32LinearTransformationwithProperties27
33TheRangeandKernel30
34MatrixRepresentationofLinearTransformation32
35Similarity36
36Reading:CramerandGershgorin38
37SelfAssessmentExercises39
Chapter4JordanCanonicalForm41
41Diagonalizability41
42JordanBlockMatrixandJordanFormMatrix44
43-matrixandSmithStandardForm45
44JordanCanonicalForm49
45Cayley-HamiltonTheoremandMinimalPolynomial53
46MATLABCommandsforJordanCanonicalForm55
47Reading:CamilleJordan57
48SelfAssessmentExercises58
Chapter5MatrixFactorization60
51Introduction60
52FullRankDecomposition60
53LUFactorization63
54QRFactorization67
55SchurDecomposition69
56SingularValueDecomposition(SVD)71
57AnExampleofApplication:ImageCompression74
58MATLABCommandsforMatrixFactorization76
59Reading:HouseholderandHouseholderAward78
510SelfAssessmentExercises80
Chapter6HermitianMatrixandPositiveDeˉniteMatrix81
61HermitianMatrix81
62PositiveDeˉniteMatrix84
63Reading:CharlesHermite87
64SelfAssessmentExercises88
Chapter7MatrixNormandMatrixAnalysis89
71Introduction89
72VectorNorm89
73MatrixNorm91
74MatrixSequence，SeriesandFunction94
75MATLABCommandsforNorms98
76Reading:Frobenius99
77SelfAssessmentExercises100
Chapter8TheMoore-PenroseGeneralizedInverse102
81Introduction102
82TheMoore-PenroseGeneralizedInverse102
83TheSolvabilityofLinearSystems105
84MATLABCommandsfortheMoore-PenroseGeneralizedInverseMatrix107
85Reading:MooreandPenrose108
86SelfAssessmentExercises109
Chapter9AnIntroductiontoMATLAB110
91AGlanceofMATLAB110
92StartingUp111
93MATLABasACalculator112
94Plotting119
95Programming122
96CommonlyUsedCommandsSummary127
97Reading:GolubandMoler130
AnswerstoSelectedExercises132
Bibliography149
Chapter 1
Review and Miscellanea: Basic Concepts in Linear Algebra
We brie°y review， mostly without proof， the basic concepts and results taught in an ele- mentary linear algebra course.
1.1 Matrix Concept and Special Matrices
The term of matrix was ˉrst introduced by the British mathematician James Joseph Sylvester in 1890. The word \matrix" is derived from the Indo-European root mater， mean- ing \mother". Matrices are indeed the core of linear algebra.
Firstly， we see the following two numbers tables(Table 1.1 and Table 1.2) in real life. Table 1.1 The price table of four kinds of cans in three supermarkets
We can write it in brief in the form
Table 1.2 The distances of three cities in China (km)
We can write it in the brief form
It is symmetric.
Deˉnition 1.1.1(Matrix) An array of numbers (or symbols) in m rows and n columns
is called an m ￡ n matrix.
The notation
or
Usually， we denote
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