Page 13 - Advanced Linear Algebra

Contents xv

Indecomposable Modules, 158
Exercises, 159
7 The Structure of a Linear Operator, 163
The Module Associated with a Linear Operator, 164
, 167
The Primary Cyclic Decomposition of =
The Characteristic Polynomial, 170
Cyclic and Indecomposable Modules, 171
The Big Picture, 174
The Rational Canonical Form, 176
Exercises, 182
8 Eigenvalues and Eigenvectors, 185
Eigenvalues and Eigenvectors, 185
Geometric and Algebraic Multiplicities, 189
The Jordan Canonical Form, 190
Triangularizability and Schur's Theorem, 192
Diagonalizable Operators, 196
Exercises, 198
9 Real and Complex Inner Product Spaces, 205
Norm and Distance, 208
Isometries, 210
Orthogonality, 211
Orthogonal and Orthonormal Sets, 212
The Projection Theorem and Best Approximations, 219
The Riesz Representation Theorem, 221
Exercises, 223
10 Structure Theory for Normal Operators, 227
The Adjoint of a Linear Operator, 227
Orthogonal Projections, 231
Unitary Diagonalizability, 233
Normal Operators, 234
Special Types of Normal Operators, 238
Self-Adjoint Operators, 239
Unitary Operators and Isometries, 240
The Structure of Normal Operators, 245
Functional Calculus, 247
Positive Operators, 250
The Polar Decomposition of an Operator, 252
Exercises, 254

Part II—Topics, 257
11 Metric Vector Spaces: The Theory of Bilinear Forms, 259
Symmetric, Skew-Symmetric and Alternate Forms, 259
The Matrix of a Bilinear Form, 261