vi Contents
Chapter 4
Polynomials 63
Degree ................................. 64
Complex Coefficients ........................ 67
Real Coefficients ........................... 69
Exercises ................................ 73
Chapter 5
Eigenvalues and Eigenvectors 75
Invariant Subspaces ......................... 76
Polynomials Applied to Operators ................. 80
Upper-Triangular Matrices ..................... 81
Diagonal Matrices ........................... 87
Invariant Subspaces on Real Vector Spaces ........... 91
Exercises ................................ 94
Chapter 6
Inner-Product Spaces 97
Inner Products ............................. 98
Norms ................................. 102
Orthonormal Bases .......................... 106
Orthogonal Projections and Minimization Problems ...... 111
Linear Functionals and Adjoints .................. 117
Exercises ................................ 122
Chapter 7
Operators on Inner-Product Spaces 127
Self-Adjoint and Normal Operators ................ 128
The Spectral Theorem ........................ 132
Normal Operators on Real Inner-Product Spaces ........ 138
Positive Operators .......................... 144
Isometries ............................... 147
Polar and Singular-Value Decompositions ............ 152
Exercises ................................ 158
Chapter 8
Operators on Complex Vector Spaces 163
Generalized Eigenvectors ...................... 164
The Characteristic Polynomial ................... 168
Decomposition of an Operator ................... 173
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