| List of exercises | p. xi |
| Preface to the series | p. xxv |
| Preface | p. xxix |
| Vectors | p. 1 |
| Real vectors | p. 4 |
| Complex vectors | p. 11 |
| Matrices | p. 15 |
| Real matrices | p. 19 |
| Complex matrices | p. 39 |
| Vector spaces | p. 43 |
| Complex and real vector spaces | p. 47 |
| Inner-product space | p. 61 |
| Hilbert space | p. 67 |
| Rank, inverse, and determinant | p. 73 |
| Rank | p. 75 |
| Inverse | p. 83 |
| Determinant | p. 87 |
| Partitioned matrices | p. 97 |
| Basic results and multiplication relations | p. 98 |
| Inverses | p. 103 |
| Determinants | p. 109 |
| Rank (in)equalities | p. 119 |
| The sweep operator | p. 126 |
| Systems of equations | p. 131 |
| Elementary matrices | p. 132 |
| Echelon matrices | p. 137 |
| Gaussian elimination | p. 143 |
| Homogeneous equations | p. 148 |
| Nonhomogeneous equations | p. 151 |
| Eigenvalues, eigenvectors, and factorizations | p. 155 |
| Eigenvalues and eigenvectors | p. 158 |
| Symmetric matrices | p. 175 |
| Some results for triangular matrices | p. 182 |
| Schur's decomposition theorem and its consequences | p. 187 |
| Jordan's decomposition theorem | p. 192 |
| Jordan chains and generalized eigenvectors | p. 201 |
| Positive (semi)definite and idempotent matrices | p. 209 |
| Positive (semi)definite matrices | p. 211 |
| Partitioning and positive (semi)definite matrices | p. 228 |
| Idempotent matrices | p. 231 |
| Matrix functions | p. 243 |
| Simple functions | p. 246 |
| Jordan representation | p. 255 |
| Matrix-polynomial representation | p. 265 |
| Kronecker product, vec-operator, and Moore-Penrose inverse | p. 273 |
| The Kronecker product | p. 274 |
| The vec-operator | p. 281 |
| The Moore-Penrose inverse | p. 284 |
| Linear vector and matrix equations | p. 292 |
| The generalized inverse | p. 295 |
| Patterned matrices: commutation- and duplication matrix | p. 299 |
| The commutation matrix | p. 300 |
| The symmetrizer matrix | p. 307 |
| The vech-operator and the duplication matrix | p. 311 |
| Linear structures | p. 318 |
| Matrix inequalities | p. 321 |
| Cauchy-Schwarz type inequalities | p. 322 |
| Positive (semi)definite matrix inequalities | p. 325 |
| Inequalities derived from the Schur complement | p. 341 |
| Inequalities concerning eigenvalues | p. 343 |
| Matrix calculus | p. 351 |
| Basic properties of differentials | p. 355 |
| Scalar functions | p. 356 |
| Vector functions | p. 360 |
| Matrix functions | p. 361 |
| The inverse | p. 364 |
| Exponential and logarithm | p. 368 |
| The determinant | p. 369 |
| Jacobians | p. 373 |
| Sensitivity analysis in regression models | p. 375 |
| The Hessian matrix | p. 378 |
| Least squares and best linear unbiased estimation | p. 382 |
| Maximum likelihood estimation | p. 387 |
| Inequalities and equalities | p. 391 |
| Some mathematical tools | p. 397 |
| Some methods of indirect proof | p. 397 |
| Primer on complex numbers and polynomials | p. 398 |
| Series expansions | p. 401 |
| Sequences and limits | p. 402 |
| Convergence of series | p. 403 |
| Special series | p. 404 |
| Expansions of functions | p. 407 |
| Multiple series, products, and their relation | p. 408 |
| Further calculus | p. 409 |
| Linear difference equations | p. 409 |
| Convexity | p. 410 |
| Constrained optimization | p. 410 |
| Notation | p. 415 |
| Vectors and matrices | p. 415 |
| Mathematical symbols, functions, and operators | p. 418 |
| Bibliography | p. 423 |
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