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! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR | ! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR | ||
|- | |- | ||
| 1 || | | 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133 | ||
|- | |- | ||
| 2 || | | 6.2 || Algebras. The Algebra of Polynomials || 136 | ||
|- | |- | ||
| 3 || | | 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142 | ||
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| 4 || | | 6.4 || Elementary Divisors || 147 | ||
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| 5 || | | 6.5 || Further Implications || 153 | ||
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| 6 || | | 6.6 || The Real Jordan Canonical Form || 155 | ||
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| 6.7 || Spectra, Jets and Polynomials || 160 | |||
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| | | 6.8 || Operator Functions and Their Matrices || 169 | ||
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| | | || Problems || 176 | ||
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! colspan="3" | | ! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS | ||
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| 7.1 || Bilinear Forms || 179 | |||
|- | |- | ||
| | | 7.2 || Quadratic Forms || 183 | ||
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| | | 7.3 || Reduction of a Quadratic Form to Canonical Form || 183 | ||
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| | | 7.4 || The Canonical Basis of a Bilinear Form || 183 | ||
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| | | 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183 | ||
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| 7.6 || Adjoint Linear Operators || 183 | |||
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| | | 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183 | ||
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| | | *7.8 || Multilinear Forms || 183 | ||
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| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183 | |||
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| | | || Problems || 210 | ||
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| | ! colspan="3" | Chapter 8: EUCLIDEAN SPACES | ||
|- | |- | ||
| | | 8.1 || Introduction || 214 | ||
|- | |- | ||
| | | 8.2 || Definition of a Euclidean Space || 215 | ||
|- | |- | ||
| 8.3 || Basic Metric Concepts || 216 | |||
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| | | 8.4 || Orthogonal Bases || 222 | ||
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| | | 8.5 || Perpendiculars || 223 | ||
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| | | 8.6 || The Orthogonalization Theorem || 226 | ||
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| | | 8.7 || The Gram Determinant || 230 | ||
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| | | 8.8 || Incompatible Systems and the Method of Least Squares || 234 | ||
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| | | 8.9 || Adjoint Operators and Isometry || 237 | ||
|- | |- | ||
| || Problems || 241 | |||
|- | |- | ||
| | ! colspan="3" | Chapter 9: UNITARY SPACES | ||
|- | |- | ||
| | | 9.1 || Hermitian Forms || 247 | ||
|- | |- | ||
| | | 9.2 || The Scalar Product in a Complex Space || 254 | ||
|- | |- | ||
| | | 9.3 || Normal Operators || 259 | ||
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| | | 9.4 || Applications to Operator Theory in Euclidean Space || 263 | ||
|- | |- | ||
| || Problems || 271 | |||
|- | |- | ||
! colspan="3" | Chapter | ! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES | ||
|- | |- | ||
| 1 || | | 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273 | ||
|- | |- | ||
| 2 || | | 10.2 || Extremal Properties of a Quadratic Form || 276 | ||
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| 3 || | | 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283 | ||
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| 4 || | | 10.4 || Reduction of the General Equation of a Quadric Surface || 287 | ||
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| 5 || | | 10.5 || Geometric Properties of a Quadric Surface || 289 | ||
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| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300 | |||
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| | | 10.7 || Hermitian Quadratic Forms || 308 | ||
|- | |- | ||
| | | || Problems || 310 | ||
|- | |- | ||
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS | |||
|- | |- | ||
| 11.1 || More on Algebras || 312 | |||
|- | |- | ||
| | | 11.2 || Representations of Abstract Algebras || 313 | ||
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| | | 11.3 || Irreducible Representations and Schur's Lemma || 314 | ||
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| 11.4 || Basic Types of Finite-Dimensional Algebras || 315 | |||
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| | | 11.5 || The Left Regular Representation of a Simple Algebra || 318 | ||
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| | | 11.6 || Structure of Simple Algebras || 320 | ||
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| | | 11.7 || Structure of Semisimple Algebras || 323 | ||
|- | |- | ||
| 11.8 || Representations of Simple and Semisimple Algebras || 327 | |||
|- | |- | ||
| | | 11.9 || Some Further Results || 331 | ||
|- | |- | ||
| | | || Problems || 332 | ||
|- | |- | ||
| | | *Appendix || || | ||
|- | |- | ||
! colspan="3" | CATEGORIES OF FINITE-DIMENSIONAL SPACES | |||
|- | |- | ||
| | | A.1 || Introduction || 335 | ||
|- | |- | ||
| | | A.2 || The Case of Complete Algebras || 338 | ||
|- | |- | ||
! colspan="2" | | | A.3 || The Case of One-Dimensional Algebras || 340 | ||
|- | |||
| A.4 || The Case of Simple Algebras || 345 | |||
|- | |||
| A.5 || The Case of Complete Algebras of Diagonal Matrices || 353 | |||
|- | |||
| A.6 || Categories and Direct Sums || 357 | |||
|- | |||
! colspan="2" | HINTS AND ANSWERS || 361 | |||
|- | |||
! colspan="2" | BIBLIOGRAPHY || 379 | |||
|- | |||
! colspan="2" | INDEX || 381 | |||
|- | |- | ||
|} | |} | ||
[[Category:Mathematics]] | [[Category:Mathematics]] |