## Linear Algebra and Linear ModelsThe main purpose of Linear Algebra and Linear Modelsis to provide a rigorous introduction to the basic aspects of the theory of linear estimation and hypothesis testing. The necessary prerequisites in matrices, multivariate normal distribution and distributions of quadratic forms are developed along the way. The book is aimed at advanced undergraduate and first-year graduate masters students taking courses in linear algebra, linear models, multivariate analysis, and design of experiments. It should also be of use to research mathematicians and statisticians as a source of standard results and problems. |

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### Contents

Vector Spaces and Matrices | 1 |

12 Vector Spaces and Subspaces | 4 |

13 Basis and Dimension | 5 |

14 Rank | 8 |

15 Orthogonality | 10 |

16 Nonsingularity | 14 |

17 Frobenius Inequality | 16 |

18 Eigenvalues and the Spectral Theorem | 18 |

38 Exercises | 73 |

39 Hints and Solutions | 76 |

Singular Values and Their Applications | 79 |

42 Extremal Representations | 81 |

43 Majorization | 84 |

44 Principal Components | 86 |

45 Canonical Correlations | 88 |

46 Volume of a Matrix | 89 |

19 Exercises | 22 |

110 Hints and Solutions | 25 |

Linear Estimation | 29 |

22 Linear Model | 33 |

23 Estimability | 35 |

24 Weighing Designs | 38 |

25 Residual Sum of Squares | 40 |

26 Estimation Subject to Restrictions | 42 |

27 Exercises | 46 |

28 Hints and Solutions | 48 |

Tests of Linear Hypotheses | 51 |

32 Multivariate Normal Distribution | 53 |

33 Quadratic Forms and Cochrans Theorem | 57 |

34 OneWay and TwoWay Classifications | 61 |

35 General Linear Hypothesis | 65 |

36 Extrema of Quadratic Forms | 67 |

37 Multiple Correlation and Regression Models | 69 |

47 Exercises | 93 |

48 Hints and Solutions | 94 |

Block Designs and Optimality | 99 |

52 The CMatrix | 102 |

53 E A and DOptimality | 103 |

54 Exercises | 108 |

55 Hints and Solutions | 110 |

Rank Additivity | 112 |

62 Characterizations of Rank Additivity | 114 |

63 General Linear Model | 118 |

64 The Star Order | 122 |

65 Exercises | 124 |

66 Hints and Solutions | 126 |

Notes | 129 |

132 | |

136 | |

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### Common terms and phrases

Algebra basis BIBD block BLUE Chapter Cochran’s theorem column rank column vector D-optimality deﬁned deﬁnition denote dim(S dispersion matrix eigenvalues eigenvector estimable Exercise F-statistic ﬁrst Frobenius Inequality GA)z given group inverse Hadamard Hadamard inequality hence Hint HintsandSolutions idempotent inequality j)-entry least squares g-inverse Let A,B Let x1 LetA linear combination linear function linear model linear span linearly independent matrices of order matrix and let matrix of rank minimum norm g-inverse model E(y Moore–Penrose inverse multiplicity n×n matrix nonsingular matrix nonzero observations orthogonal matrix orthonormal partitioned positive definite positive semidefinite positive semidefinite matrix principal component principal minors principal submatrix proof is complete R(AB rank factorization real numbers row space Schur complement Similarly singular values solution spectral theorem square matrix star order submatrix subspace Suppose symmetric matrix treatment unbiased variance VC1V vector space vectors x1 zero