Introduction p. 1
The scope of matrix algebra p. 1
General description of a matrix p. 3
Subscript notation p. 4
Summation notation p. 6
Dot notation p. 11
Definition of a matrix p. 12
Vectors and scalars p. 15
General notation p. 16
Illustrative examples p. 16
Exercises p. 17
Basic Operations p. 22
The transpose of a matrix p. 22
A reflexive operation p. 23
Vectors p. 24
Partitioned matrices p. 24
Example p. 24
General specification p. 26
Transposing a partitioned matrix p. 26
Partitioning into vectors p. 26
The trace of a matrix p. 27
Addition p. 28
Scalar multiplication p. 29
Subtraction p. 30
Equality and the null matrix p. 31
Multiplication p. 32
The inner product of two vectors p. 32
A matrix-vector product p. 33
A product of two matrices p. 36
Existence of matrix products p. 38
Products with vectors p. 39
Products with scalars p. 42
Products with null matrices p. 43
Products with diagonal matrices p. 43
Identity matrices p. 44
The transpose of a product p. 44
The trace of a product p. 45
Powers of a matrix p. 46
Partitioned matrices p. 48
Hadamard products p. 49
The Laws of algebra p. 50
Associative laws p. 50
The distributive law p. 51
Commutative laws p. 51
Contrasts with scalar algebra p. 52
Exercises p. 53
Special Matrices p. 60
Symmetric matrices p. 60
Products of symmetric matrices p. 61
Properties of AA' and A'A p. 61
Products of vectors p. 63
Sums of outer products p. 64
Elementary vectors p. 65
Skew-symmetric matrices p. 65
Matrices having all elements equal p. 65
Idempotent matrices p. 68
Orthogonal matrices p. 69
Definitions p. 69
Special cases p. 71
Quadratic forms p. 73
Positive definite matrices p. 76
Exercises p. 78
Determinants p. 84
Expansion by minors p. 84
First- and second-order determinants p. 85
Third-order determinants p. 86
n-order determinants p. 89
Formal definition p. 90
Basic properties p. 92
Determinant of a transpose p. 92
Two rows the same p. 93
Cofactors p. 93
Adding multiples of a row (column) to a row (column) p. 95
Products p. 96
Elementary row operations p. 99
Factorization p. 101
A row (column) of zeros p. 102
Interchanging rows (columns) p. 102
Adding a row to a multiple of a row p. 102
Examples p. 103
Diagonal expansion p. 106
The Laplace expansion p. 109
Sums and differences of determinants p. 111
Exercises p. 112
Inverse Matrices p. 119
Introduction: solving equations p. 119
Products equal to I p. 123
Cofactors of a determinant p. 125
Derivation of the inverse p. 125
Conditions for existence of the inverse p. 129
Properties of the inverse p. 130
Some simple special cases p. 131
Inverses of order 2 p. 131
Diagonal matrices p. 132
I and J matrices p. 132
Orthogonal matrices p. 132
Idempotent matrices p. 133
Equations and algebra p. 133
Solving linear equations p. 133
Algebraic simplifications p. 136
Computers and inverses p. 139
The arithmetic of linear equations p. 140
Rounding error p. 142
Left and right inverses p. 147
Exercises p. 148
Rank p. 155
Linear combinations of vectors p. 155
Linear transformations p. 157
Linear dependence and independence p. 159
Definitions p. 159
General characteristics p. 161
Linearly dependent vectors p. 162
At least two a's are nonzero p. 162
Vectors are linear combinations of others p. 162
Partitioning matrices p. 163
Zero determinants p. 164
Inverse matrices p. 164
Testing for dependence (simple cases) p. 164
Linearly independent (LIN) vectors p. 166
Nonzero determinants and inverse matrices p. 166
Linear combinations of LIN vectors p. 167
A maximum number of LIN vectors p. 167
The number of LIN rows and columns in a matrix p. 169
The rank of a matrix p. 171
Rank and inverse matrices p. 172
Permutation matrices p. 173
Full-rank factorization p. 175
Basic development p. 175
The general case p. 177
Matrices of full row (column) rank p. 177
Vector spaces p. 177
Euclidean space p. 178
Vector spaces p. 178
Spanning sets and bases p. 179
Many spaces of order n p. 179
Subspaces p. 180
The range and null space of a matrix p. 180
Exercises p. 181
Canonical Forms p. 184
Elementary operators p. 184
Row operations p. 184
Transposes p. 185
Column operations p. 185
Inverses p. 185
Rank and the elementary operators p. 186
Rank p. 186
Products of elementary operators p. 186
Equivalence p. 187
Finding the rank of a matrix p. 187
Some special LIN vectors p. 187
Calculating rank p. 188
A general procedure p. 189
Reduction to equivalent canonical form p. 190
Row operations p. 190
Column operations p. 191
The equivalent canonical form p. 192
Non-uniqueness of P and Q p. 194
Existence is assured p. 194
Full-rank factorization p. 194
Rank of a product matrix p. 196
Symmetric matrices p. 199
Row and column operations p. 199
The diagonal form p. 200
The canonical form under congruence p. 201
Two special provisions p. 202
Full-rank factorization p. 205
Non-negative definite matrices p. 205
Diagonal elements and principal minors p. 205
Congruent canonical form p. 206
Full-rank factorization p. 206
Quadratic forms as sums of squares p. 206
Full row (column) rank matrices p. 208
Exercises p. 209
Generalized Inverses p. 212
The Moore-Penrose inverse p. 212
Generalized inverses p. 212
Derivation from row operations p. 214
Derivation from the diagonal form p. 215
Other names and symbols p. 216
An algorithm p. 217
An easy form p. 217
A general form p. 217
Arbitrariness in a generalized inverse p. 219
Symmetric matrices p. 220
Non-negative definite matrices p. 220
A general algorithm p. 221
The matrix X'X p. 221
Exercises p. 222
Solving Linear Equations p. 227
Equations having many solutions p. 227
Consistent equations p. 228
Definition p. 228
Existence of solutions p. 229
Tests for consistency p. 232
Equations having one solution p. 233
Deriving solutions using generalized inverses p. 235
Obtaining a solution p. 235
Obtaining many solutions p. 236
All possible solutions p. 237
Combinations of solutions p. 239
Linearly independent solutions p. 239
An invariance property p. 242
Equations Ax = 0 p. 245
General properties p. 245
Orthogonal solutions p. 246
Orthogonal vector spaces p. 248
A complete example p. 249
Least squares equations p. 251
Exercises p. 252
Partitioned Matrices p. 257
Orthogonal matrices p. 257
Determinants p. 258
Inverses p. 260
Schur complements p. 261
Generalized inverses p. 261
Direct sums p. 263
Direct products p. 265
Exercises p. 267
Eigenvalues and Eigenvectors p. 272
Introduction: age distribution vectors p. 272
Derivation of eigenvalues p. 274
Elementary properties of eigenvalues p. 276
Eigenvalues of powers of a matrix p. 276
Eigenvalues of a scalar-by-matrix product p. 277
Eigenvalues of polynomials p. 277
The sum and product of eigenvalues p. 278
Calculating eigenvectors p. 279
A general method p. 279
Simple roots p. 280
Multiple roots p. 281
The similar canonical form p. 282
Derivation p. 282
Uses p. 285
Symmetric matrices p. 290
Eigenvalues all real p. 290
Symmetric matrices are diagonable p. 290
Eigenvectors are orthogonal p. 290
Rank equals number of nonzero eigenvalues p. 292
Dominant eigenvalues p. 293
Factoring the characteristic equation p. 298
Exercises p. 299
Appendix to Chapter 11 p. 305
Proving the diagonability theorem p. 305
The number of nonzero eigenvalues never exceeds rank p. 305
A lower bound on r(A - [lambda subscript k]I) p. 306
Proof of the diagonability theorem p. 307
All symmetric matrices are diagonable p. 307
Other results for symmetric matrices p. 308
Spectral decomposition p. 308
Non-negative definite (n.n.d.) matrices p. 309
Simultaneous diagonalization of two symmetric matrices p. 312
The Cayley-Hamilton theorem p. 314
The singular-value decomposition p. 316
Exercises p. 318
Miscellanea p. 320
Orthogonal matrices-a summary p. 320
Idempotent matrices-a summary p. 320
The matrix aI + bJ-a summary p. 322
Non-negative definite matrices-a summary p. 322
Canonical forms and other decompositions-a summary p. 323
Matrix Functions p. 325
Functions of matrices p. 325
Matrices of functions p. 325
Iterative solution of nonlinear equations p. 326
Vectors of differential operators p. 327
Scalars p. 327
Vectors p. 328
Quadratic forms p. 329
Vec and vech operators p. 332
Definitions p. 332
Properties of vec p. 333
Vec-permutation matrices p. 334
Relationships between vec and vech p. 334
Other calculus results p. 334
Differentiating inverses p. 335
Differentiating traces p. 335
Differentiating determinants p. 336
Jacobians p. 338
Aitken's integral p. 340
Hessians p. 341
Matrices with elements that are complex numbers p. 341
Exercises p. 342
Applications in Statistics p. 346
Variance-covariance matrices p. 347
Correlation matrices p. 348
Matrices of sums of squares and cross-products p. 349
Data matrices p. 349
Uncorrected sums of squares and products p. 349
Means, and the centering matrix p. 350
Corrected sums of squares and products p. 351
The multivariate normal distribution p. 352
Quadratic forms and X[superscript 2]-distributions p. 355
Least squares equations p. 357
Contrasts among means p. 358
Exercises p. 360
The Matrix Algebra of Regression Analysis p. 363
General description p. 363
Linear models p. 364
Observations p. 364
Nonlinear models p. 365
Estimation p. 366
Several regressor variables p. 368
Deviations from means p. 369
The statistical model p. 371
Unbiasedness and variances p. 372
Predicted y-values p. 373
Estimating the error variance p. 374
Partitioning the total sum of squares p. 376
Multiple correlation p. 378
The no-intercept model p. 379
Analysis of variance p. 380
Testing linear hypotheses p. 382
Stating a hypothesis p. 382
The F-statistic p. 383
Equivalent statements of a hypothesis p. 385
Special cases p. 386
Confidence intervals p. 386
Fitting subsets of the x-variables p. 387
Reductions in sums of squares: the R([characters not reproducible]) notation p. 389
An Introduction to Linear Statistical Models p. 392
General description p. 392
The normal equations p. 395
A general form p. 395
Many solutions p. 396
Solving the normal equations p. 397
Generalized inverses of X'X p. 397
Solutions p. 397
Expected values and variances p. 399
Predicted y-values p. 400
Estimating the error variance p. 401
Error sum of squares p. 401
Expected value p. 402
Estimation p. 402
Partitioning the total sum of squares p. 403
Coefficient of determination p. 404
Analysis of variance p. 405
The R([characters not reproducible]) notation p. 407
Estimable functions p. 408
Testing linear hypotheses p. 411
Confidence intervals p. 415
Some particular models p. 416
The one-way classification p. 416
Two-way classification, no interactions, balanced data p. 418
Two-way classification, no interactions, unbalanced data p. 422
The R([characters not reproducible]) notation (Continued) p. 424
References p. 429
Index p. 433