1 Algebraic Preliminaries ...................................... 1
1.1 Introduction ............................................ 1
1.2 Sets and Maps ........................................... 1
1.3 Groups .................................................. 3
1.4 Rings and Fields ........................................ 8
1.4.1 The Integers .................................... 11
1.4.2 The Polynomial Ring ............................. 11
1.4.3 Formal Power Series ............................. 20
1.4.4 Rational Functions .............................. 22
1.4.5 Proper Rational Functions ....................... 23
1.4.6 Stable Rational Functions ....................... 24
1.4.7 Truncated Laurent Series ........................ 25
1.5 Modules ................................................ 26
1.6 Exercises .............................................. 31
1.7 Notes and Remarks ...................................... 32
2 Vector Spaces ............................................... 33
2.1 Introduction ........................................... 33
2.2 Vector Spaces .......................................... 33
2.3 Linear Combinations .................................... 36
2.4 Subspaces .............................................. 36
2.5 Linear Dependence and Independence ..................... 37
2.6 Subspaces and Bases .................................... 40
2.7 Direct Sums ............................................ 41
2.8 Quotient Spaces ........................................ 43
2.9 Coordinates ............................................ 45
2.10 Change of Basis Transformations ........................ 46
2.11 Lagrange Interpolation ................................. 48
2.12 Taylor Expansion ....................................... 51
2.13 Exercises .............................................. 52
2.14 Notes and Remarks ...................................... 52
3 Determinants ................................................ 55
3.1 Introduction ........................................... 55
3.2 Basic Properties ....................................... 55
3.3 Cramer's Rule .......................................... 60
3.4 The Sylvester Resultant ................................ 62
3.5 Exercises .............................................. 64
3.6 Notes and Remarks ...................................... 66
4 Linear Transformations ...................................... 67
4.1 Introduction ........................................... 67
4.2 Linear Transformations ................................. 67
4.3 Matrix Representations ................................. 75
4.4 Linear Functionals and Duality ......................... 79
4.5 The Adjoint Transformation ............................. 85
4.6 Polynomial Module Structure on Vector Spaces ........... 88
4.7 Exercises .............................................. 94
4.8 Notes and Remarks ...................................... 95
5 The Shift Operator .......................................... 97
5.1 Introduction ........................................... 97
5.2 Basic Properties ....................................... 97
5.3 Circulant Matrices .................................... 109
5.4 Rational Models ....................................... 111
5.5 The Chinese Remainder Theorem and Interpolation ....... 118
5.5.1 Lagrange Interpolation Revisited ............... 119
5.5.2 Hermite Interpolation .......................... 120
5.5.3 Newton Interpolation ........................... 121
5.6 Duality ............................................... 122
5.7 Universality of Shifts ................................ 127
5.8 Exercises ............................................. 131
5.9 Notes and Remarks ..................................... 133
6 Structure Theory of Linear Transformations ................. 135
6.1 Introduction .......................................... 135
6.2 Cyclic Transformations ................................ 135
6.2.1 Canonical Forms for Cyclic Transformations ..... 140
6.3 The Invariant Factor Algorithm ........................ 145
6.4 Noncyclic Transformations ............................. 147
6.5 Diagonalization ....................................... 151
6.6 Exercises ............................................. 154
6.7 Notes and Remarks ..................................... 158
7 Inner Product Spaces ....................................... 161
7.1 Introduction .......................................... 161
7.2 Geometry of Inner Product Spaces ...................... 161
7.3 Operators in Inner Product Spaces ..................... 166
7.3.1 The Adjoint Transformation ..................... 166
7.3.2 Unitary Operators .............................. 169
7.3.3 Self-adjoint Operators ......................... 173
7.3.4 The Minimax Principle .......................... 176
7.3.5 The Cayley Transform ........................... 176
7.3.6 Normal Operators ............................... 178
7.3.7 Positive Operators ............................. 180
7.3.8 Partial Isometries ............................. 182
7.3.9 The Polar Decomposition ........................ 183
7.4 Singular Vectors and Singular Values .................. 184
7.5 Unitary Embeddings .................................... 187
7.6 Exercises ............................................. 190
7.7 Notes and Remarks ..................................... 193
8 Tensor Products and Forms .................................. 195
8.1 Introduction .......................................... 195
8.2 Basics ................................................ 196
8.2.1 Forms in Inner Product Spaces .................. 196
8.2.2 Sylvester's Law of Inertia ..................... 199
8.3 Some Classes of Forms ................................. 204
8.3.1 Hankel Forms ................................... 205
8.3.2 Bezoutians ..................................... 209
8.3.3 Representation of Bezoutians ................... 213
8.3.4 Diagonalization of Bezoutians .................. 217
8.3.5 Bezout and Hankel Matrices ..................... 223
8.3.6 Inversion of Hankel Matrices ................... 230
8.3.7 Continued Fractions and Orthogonal
Polynomials .................................... 237
8.3.8 The Cauchy Index ............................... 248
8.4 Tensor Products of Models ............................. 254
8.4.1 Bilinear Forms ................................. 254
8.4.2 Tensor Products of Vector Spaces ............... 255
8.4.3 Tensor Products of Modules ..................... 259
8.4.4 Kronecker Product Models ....................... 260
8.4.5 Tensor Products over a Field ................... 261
8.4.6 Tensor Products over the Ring of Polynomials ... 263
8.4.7 The Polynomial Sylvester Equation .............. 266
8.4.8 Reproducing Kernels ............................ 270
8.4.9 The Bezout Map ................................. 272
8.5 Exercises ............................................. 274
8.6 Notes and Remarks ..................................... 277
9 Stability .................................................. 279
9.1 Introduction .......................................... 279
9.2 Root Location Using Quadratic Forms ................... 279
9.3 Exercises ............................................. 293
9.4 Notes and Remarks ..................................... 294
10 Elements of Linear System Theory ........................... 295
10.1 Introduction ..........................................
10 2 Systems and Their Representations ..................... 296
10.3 Realization Theory ....................................
10.4 Stabilization ......................................... 314
10 5 The Youla-Kucera Parametrization ...................... 319
10.6 Exercises ............................................. 321
10.7 Notes and Remarks ..................................... 324
11 Rational Hardy Spaces ......................................
11.1 Introduction .......................................... 325
11.2 Hardy Spaces and Their Maps ........................... 326
11.2.1 Rational Hardy Spaces .......................... 326
11.2.2 Invariant Subspaces ............................ 334
11.2.3 Model Operators and Intertwining Maps .......... 337
11.2.4 Intertwining Maps and Interpolation ............ 344
11.2.5 RH+-Chinese Remainder Theorem .................. 353
11.2.6 Analytic Hankel Operators and Intertwining
Maps ........................................... 354
11.3 Exercises ............................................. JO
11.4 Notes and Remarks .....................................
12 Model Reduction ............................................ 361
12.1 Introduction .......................................... 361
12.2 Hankel Norm Approximation ............................. 362
12.2.1 Schmidt Pairs of Hankel Operators .............. 363
12.2.2 Reduction to Eigenvalue Equation ............... 369
12.2.3 Zeros of Singular Vectors and a Bezout
equation ....................................... 370
12.2.4 More on Zeros of Singular Vectors .............. 376
12.2.5 Nehari's Theorem ............................... 378
12.2.6 Nevanlinna-Pick Interpolation .................. 379
12.2.7 Hankel Approximant Singular Values and
Vectors ........................................ 381
12.2.8 Orthogonality Relations ........................ 383
12.2.9 Duality in Hankel Norm Approximation ........... 385
12.3 Model Reduction: A Circle of Ideas .................... 392
12.3.1 The Sylvester Equation and Interpolation ....... 392
12 3 2 The Sylvester Equation and the Projection
Method ......................................... 394
12.4 Exercises .............................................
12.5 Notes ................................................. 400
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