Preface ........................................................ xi
Introduction..................................................... 1
Part I. Motivating Problems, Systems and Realizations
1. Motivating Problems
1.1 Linear time invariant systems and cascade connection ..... 7
1.2 Characteristic operator functions and invariant
subspaces (1) ........................................... 11
1.3. Characteristic operator functions and invariant
subspaces (2) ........................................... 14
1.4. Factorization of monic matrix polynomials .............. 17
1.5. Wiener-Hopf integral operators and factorization ....... 18
1.6. Block Toeplitz equations and factorization ............. 21
Notes ................................................... 23
2. Operator Nodes, Systems, and Operations on Systems
2.1. Operator nodes, systems and transfer functions ......... 25
2.2. Inversion .............................................. 27
2.3. Products ............................................... 30
2.4. Factorization and matching of invariant subspaces ...... 32
2.5. Factorization and inversion revisited .................. 37
Notes .................................................. 48
3. Various Classes of Systems
3.1. Brodskii systems ....................................... 49
3.2. Krein systems .......................................... 50
3.3. Unitary systems ........................................ 51
3.4. Monic systems .......................................... 53
3.5. Polynomial systems ..................................... 57
3.6. Mobius transformation of systems ....................... 58
Notes .................................................. 64
4. Realization and Linearization of Operator Functions
4.1. Realization of rational operator functions ............. 65
4.2. Realization of analytic operator functions ............. 67
4.3. Linearization .......................................... 69
4.4. Linearization and Schur complements .................... 73
Notes .................................................. 76
5. Factorization and Riccati Equations
5.1. Angular subspaces and angular operators ................ 77
5.2. Angular subspaces and the algebraic Riccati equation ... 79
5.3. Angular operators and factorization .................... 80
5.4. Angular spectral subspaces and the algebraic Riccati
equation ............................................... 86
Notes .................................................. 88
6. Canonical Factorization and Applications
6.1. Canonical factorization of rational matrix
functions .............................................. 89
6.2. Application to Wiener-Hopf integral equations .......... 92
6.3. Application to block Toeplitz operators ................ 97
Notes ................................................. 100
Part II. Minimal Realization and Minimal Factorization
7. Minimal Systems
7.1. Minimality of systems ................................. 105
7.2. Controllability and observability for finite-dimensional
systemsm .............................................. 109
7.3. Minimality for finite-dimensional systems ............. 112
7.4. Minimality for Hilbert space systems .................. 116
7.5. Minimality in special cases ........................... 125
7.5.1. Brodskii systems ............................... 125
7.5.2. Krein systems .................................. 125
7.5.3. Unitary systems ................................ 126
7.5.4. Monic systems .................................. 127
7.5.5. Polynomial systems ............................. 128
Notes ................................................. 128
8. Minimal Realizations and Pole-Zero Structure
8.1. Zero data and Jordan chains ........................... 129
8.2. Pole data ............................................. 142
8.3. Minimal realizations in terms of zero or pole data .... 145
8.4. Local degree and local minimality ..................... 147
8.5. McMillan degree and minimality of systems ............. 160
Notes ................................................. 161
9. Minimal Factorization of Rational Matrix Functions
9.1. Minimal factorization ................................. 163
9.2. Pseudo-canonical factorization ........................ 169
9.3. Minimal factorization in a singular case .............. 172
Notes ................................................. 179
Part III. Degree One Factors, Companion Based Rational Matrix
Functions, and Job Scheduling
10. Factorization into Degree One Factors
10.1. Simultaneous reduction to complementary triangular
forms ............................................... 184
10.2. Factorization into elementary factors and
realization ......................................... 188
10.3. Complete factorization (general) .................... 195
10.4. Quasicomplete factorization (general) ............... 199
Notes ............................................... 209
11. Complete Factorization of Companion Based Matrix Functions
11.1. Companion matrices: preliminaries ................... 212
11.2. Simultaneous reduction to complementary triangular
forms ............................................... 216
11.3. Preliminaries about companion based matrix
functions ........................................... 231
11.4. Companion based matrix functions: poles and zeros ... 234
11.5. Complete factorization (companion based) ............ 244
11.6. Maple procedures for calculating complete
factorizations ...................................... 246
11.6.1. Maple environment and procedures ............ 247
11.6.2. Poles, zeros and orderings .................. 247
11.6.3. Triangularization routines (complete) ....... 251
11.6.4. Factorization procedures .................... 252
11.6.5. Example ..................................... 254
11.7 Appendix: invariant subspaces of companion
matrices ............................................ 260
Notes ............................................... 266
12. Quasicomplete Factorization and Job Scheduling
12.1. A combinatorial lemma ............................... 268
12.2. Quasicomplete factorization (companion based) ....... 272
12.3. A review of the two machine flow shop problem ....... 288
12.4. Quasicomplete factorization and the 2MSFP ........... 293
12.5. Maple procedures for quasicomplete factorizations ... 301
12.5.1. Maple environment ........................... 302
12.5.2. Triangularization routines
(quasicomplete) ............................. 303
12.5.3. Transformations into upper triangular
form ........................................ 307
12.5.4. Transformation into complementary triangular
forms ....................................... 308
12.5.5. An example: symbolic and quasicomplete ...... 309
12.5.6. Concluding remarks .......................... 314
Notes ............................................... 315
Part IV. Stability of Factorization and of Invariant Subspaces
13. Stability of Spectral Divisors
13.1. Examples and first results for the finite-dimensional
case ................................................ 319
13.2. Opening between subspaces and angular operators ..... 322
13.3. Stability of spectral divisors of systems ........... 327
13.4. Applications to transfer functions .................. 332
13.5. Applications to Riccati equations ................... 335
Notes ............................................... 338
14. Stability of Divisors
14.1. Stable invariant subspaces .......................... 339
14.2. Lipschitz stable invariant subspaces ................ 345
14.3. Stable minimal factorizations of rational matrix
functions ........................................... 348
14.4. Stable complete factorizations ...................... 352
14.5. Stable factorizations of monic matrix polynomials ... 356
14.6. Stable solutions of the operator Riccati equation ... 359
14.7. Stability of stable factorizations .................. 360
14.8. Isolated factorizations and related topics .......... 363
14.8.1. Isolated invariant subspaces ................ 363
14.8.2. Isolated chains of invariant subspaces ...... 366
14.8.3. Isolated factorizations ..................... 369
14.8.4. Isolated solutions of the Riccati
equation .................................... 372
Notes ............................................... 372
15. Factorization of Real Matrix Functions
15.1. Real matrix functions ............................... 375
15.2. Real monic matrix polynomials ....................... 378
15.3. Stable and isolated invariant subspaces ............. 379
15.4. Stable and isolated real factorizations ............. 385
15.5. Stability of stable real factorizations ............. 389
Notes ............................................... 391
Bibliography .................................................. 393
List of Symbols ............................................... 401
Index ......................................................... 405
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