## Matrix Algebra From a Statistician's PerspectiveMatrix algebra plays a very important role in statistics and in many other dis- plines. In many areas of statistics, it has become routine to use matrix algebra in thepresentationandthederivationorveri?cationofresults. Onesuchareaislinear statistical models; another is multivariate analysis. In these areas, a knowledge of matrix algebra isneeded in applying important concepts, as well as instudying the underlying theory, and is even needed to use various software packages (if they are to be used with con?dence and competence). On many occasions, I have taught graduate-level courses in linear statistical models. Typically, the prerequisites for such courses include an introductory (- dergraduate) course in matrix (or linear) algebra. Also typically, the preparation provided by this prerequisite course is not fully adequate. There are several r- sons for this. The level of abstraction or generality in the matrix (or linear) algebra course may have been so high that it did not lead to a “working knowledge” of the subject, or, at the other extreme, the course may have emphasized computations at the expense of fundamental concepts. Further, the content of introductory courses on matrix (or linear) algebra varies widely from institution to institution and from instructor to instructor. Topics such as quadratic forms, partitioned matrices, and generalized inverses that play an important role in the study of linear statistical models may be covered inadequately if at all. |

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### Inhalt

Matrices | 1 |

12 Basic Operations | 2 |

13 Some Basic Types of Matrices | 5 |

Exercises | 10 |

Submatrices and Partitioned Matrices | 13 |

23 Some Results on the Product of a Matrix and a Column Vector | 19 |

24 Expansion of a Matrix in Terms of Its Rows Columns or Elements | 20 |

Linear Dependence and Independence | 23 |

154 Differentiation of Matrix Sums Products and Transposes and of Matrices of Constants | 300 |

155 Differentiation of a Vector or Unrestricted or Symmetric Matrix With Respect to Its Elements | 303 |

156 Differentiation of a Trace of a Matrix | 304 |

157 The Chain Rule | 306 |

158 FirstOrder Partial Derivatives of Determinants and Inverse and Adjoint Matrices | 308 |

159 SecondOrder Partial Derivatives of Determinants and Inverse Matrices | 312 |

1510 Differentiation of Generalized Inverses | 314 |

1511 Differentiation of Projection Matrices | 319 |

Linear Spaces Row and Column Spaces | 27 |

42 Subspaces | 29 |

43 Bases | 31 |

44 Rank of a Matrix | 36 |

45 Some Basic Results on Partitioned Matrices and on Sums of Matrices | 41 |

Exercises Section 41 | 46 |

Trace of a Square Matrix | 49 |

52 Trace of a Product | 50 |

53 Some Equivalent Conditions | 52 |

Geometrical Considerations | 54 |

62 Orthogonal and Orthonormal Sets | 61 |

63 Schwarz Inequality | 62 |

64 Orthonormal Bases | 63 |

Exercises | 68 |

Linear Systems Consistency and Compatibility | 71 |

72 Consistency | 72 |

73 Compatibility | 73 |

Exercise | 77 |

Inverse Matrices | 79 |

82 Properties of Inverse Matrices | 81 |

83 Premultiplication or Postmultiplication by a Matrix of Full Column or Row Rank | 82 |

84 Orthogonal Matrices | 84 |

85 Some Basic Results on the Ranks and Inverses of Partitioned Matrices | 88 |

Exercises | 103 |

Generalized Inverses | 106 |

92 Some Alternative Characterizations | 109 |

93 Some Elementary Properties | 117 |

94 Invariance to the Choice of a Generalized Inverse | 119 |

95 A Necessary and Sufﬁcient Condition for the Consistency of aLinearSystem | 120 |

96 Some Results on the Ranks and Generalized Inverses of Partitioned Matrices | 121 |

97 Extension of Some Results on Systems of the Form AX D B to Systems of the Form AXC D B | 126 |

Idempotent Matrices | 133 |

102 Some Basic Results | 134 |

Exercises | 136 |

Linear Systems Solutions | 139 |

112 General Form of a Solution a Homogeneous linear systems | 140 |

113 Number of Solutions | 142 |

115 An Alternative Expression for the General Form of a Solution | 144 |

116 Equivalent Linear Systems | 145 |

117 Null and Column Spaces of Idempotent Matrices | 146 |

119 A Computational Approach | 149 |

1110 Linear Combinations of the Unknowns | 150 |

1111 Absorption | 152 |

1112 Extensions to Systems of the Form AXC D B | 157 |

Projections and Projection Matrices | 161 |

122 Projection of a Column Vector | 163 |

123 Projection Matrices | 166 |

125 Orthogonal Complements | 172 |

Determinants | 179 |

132 Some Basic Properties of Determinants | 183 |

133 Partitioned Matrices Products of Matrices and Inverse Matrices | 187 |

134 A Computational Approach | 191 |

136 Vandermonde Matrices | 195 |

137 Some Results on the Determinant of the Sum of Two Matrices | 197 |

138 Laplaces Theorem and the BinetCauchy Formula | 200 |

Exercises Section 131 | 205 |

Linear Bilinear and Quadratic Forms | 209 |

142 Nonnegative Deﬁnite Quadratic Forms and Matrices a Deﬁnitions | 212 |

143 Decomposition of Symmetric and Symmetric Nonnegative Deﬁnite Matrices | 218 |

144 Generalized Inverses of Symmetric Nonnegative Deﬁnite Matrices | 222 |

145 LDU U0DU and Cholesky Decompositions | 223 |

146 SkewSymmetric Matrices | 239 |

147 Trace of a Nonnegative Deﬁnite Matrix a Basic results | 240 |

148 Partitioned Nonnegative Deﬁnite Matrices | 243 |

149 Some Results on Determinants | 247 |

1410 Geometrical Considerations | 255 |

1411 Some Results on Ranks and Row and Column Spaces and on Linear Systems | 259 |

1412 Projections Projection Matrices and Orthogonal Complements | 260 |

Exercises Section 141 | 277 |

Matrix Differentiation | 289 |

151 Deﬁnitions Notation and Other Preliminaries | 290 |

Some Elementary Results | 296 |

153 Differentiation of Linear and Quadratic Forms | 298 |

1512 Evaluation of Some Multiple Integrals | 324 |

Exercises Section 151 | 327 |

Bibliographic and Supplementary Notes | 335 |

Kronecker Products and the Vec and Vech Operators | 336 |

Definition and Some Basic Properties | 343 |

163 VecPermutation Matrix | 347 |

164 The Vech Operator | 354 |

165 Reformulation of a Linear System | 367 |

166 Some Results on Jacobian Matrices | 368 |

Exercises | 371 |

Bibliographic and Supplementary Notes | 377 |

Intersections and Sums of Subspaces | 379 |

172 Some Results on Row and Column Spaces and on the Ranks of Partitioned Matrices | 385 |

173 Some Results on Linear Systems and on Generalized Inverses of Partitioned Matrices | 392 |

Sum of Their Dimensions Versus Dimension of Their Sum | 396 |

175 Some Results on the Rank of a Product of Matrices a Initial presentation of results | 398 |

176 Projections Along a Subspace a Some general results and terminology | 402 |

177 Some Further Results on the Essential Disjointness and Orthogonality of Subspaces and on Projections and Projection Matrices | 409 |

Exercises | 411 |

Bibliographic and Supplementary Notes | 417 |

Sums and Differences of Matrices | 419 |

182 Some Results on Inverses and Generalized Inverses and on Linear Systems | 423 |

183 Some Results on Positive and Nonnegative Deﬁniteness | 437 |

184 Some Results on Idempotency a Basic results | 439 |

185 Some Results on Ranks | 444 |

Exercises | 450 |

Bibliographic and Supplementary Notes | 458 |

Minimization of a SecondDegree Polynomial in n Variables Subject to Linear Constraints | 459 |

191 Unconstrained Minimization | 460 |

192 Constrained Minimization a Basic results | 463 |

193 Explicit Expressions for Solutions to the Constrained Minimization Problem | 468 |

194 Some Results on Generalized Inverses of Partitioned Matrices | 476 |

195 Some Additional Results on the Form of Solutions to the Constrained Minimization Problem | 483 |

196 Transformation of the Constrained Minimization Problem to an Unconstrained Minimization Problem | 489 |

197 The Effect of Constraints on the Generalized Least Squares Problem | 491 |

Bibliographic and Supplementary Notes | 495 |

The MoorePenrose Inverse | 496 |

202 Some Special Cases | 499 |

203 Special Types of Generalized Inverses | 500 |

204 Some Alternative Representations and Characterizations | 507 |

205 Some Basic Properties and Relationships | 508 |

206 Minimum Norm Solution to the Least Squares Problem | 512 |

208 Differentiation of the MoorePenrose Inverse | 514 |

Exercises | 517 |

Bibliographic and Supplementary Notes | 519 |

Eigenvalues and Eigenvectors | 521 |

211 Deﬁnitions Terminology and Some Basic Results | 522 |

212 Eigenvalues of Triangular or BlockTriangular Matrices and of Diagonal or BlockDiagonal Matrices | 528 |

213 Similar Matrices | 530 |

214 Linear Independence of Eigenvectors | 534 |

215 Diagonalization | 537 |

216 Expressions for the Trace and Determinant of a Matrix | 545 |

217 Some Results on the MoorePenrose Inverse of a Symmetric Matrix | 546 |

218 Eigenvalues of Orthogonal Idempotent and Nonnegative Deﬁnite Matrices | 548 |

219 Square Root of a Symmetric Nonnegative Deﬁnite Matrix | 550 |

2110 Some Relationships | 551 |

2111 Eigenvalues and Eigenvectors of Kronecker Products of Square Matrices | 554 |

2112 Singular Value Decomposition | 556 |

2113 Simultaneous Diagonalization | 566 |

2114 Generalized Eigenvalue Problem | 569 |

2115 Differentiation of Eigenvalues and Eigenvectors | 571 |

2116 An Equivalence Involving Determinants and Polynomials | 574 |

Some Properties of Polynomials in a Single Variable | 580 |

Bibliographic and Supplementary Notes | 588 |

Linear Transformations | 589 |

224 Matrix Representation of a Linear Transformation a Background deﬁnition and some basic properties | 601 |

225 Terminology and Properties Shared by a Linear Transformation and Its Matrix Representation | 609 |

226 Linear Functionals and Dual Transformations | 612 |

Exercises Section 221 | 616 |

621 | |

624 | |

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ˇ ˇ ˇ ÃÂ AX D B column vector continuously differentiable deﬁned diagonal elements diagonal matrix eigenvalues eigenvectors equivalently essentially disjoint exists ﬁnd ﬁrst following lemma following theorem follows from Theorem formula function hence idempotent ijth element inner product interior point Kronecker product LDU decomposition Let A represent light of Corollary light of Lemma light of Theorem linear space linear system linear system AX linear transformation linearly independent m x n matrix representation Moreover n n matrix n n symmetric matrix n n symmetric nonnegative n x n n-dimensional column vector nonnegative deﬁnite matrix nonnull nonsingular matrix orthogonal matrix partitioned matrix permutation polynomial positive deﬁnite matrix positive semideﬁnite projection matrix Proof rank.A represent an n n respect satisﬁed scalar Schur complement Section Show subspace sufﬁcient Suppose symmetric matrix symmetric nonnegative deﬁnite symmetric positive deﬁnite U0DU x0Ax