## 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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### Contents

Matrices | 1 |

Submatrices and Partitioned Matrices | 13 |

Linear Dependence and Independence | 23 |

Trace of a Square Matrix | 49 |

Consistency and Compatibility | 71 |

Inverse Matrices | 79 |

Generalized Inverses 107 | 106 |

Idempotent Matrices | 133 |

Linear Bilinear and Quadratic Forms | 209 |

Matrix Differentiation | 289 |

Kronecker Products and the Vec and Vech Operators 337 | 336 |

Intersections and Sums of Subspaces | 379 |

Sums and Differences of Matrices | 419 |

Minimization of a SecondDegree Polynomial in n Variables | 459 |

The MoorePenrose Inverse 497 | 496 |

Eigenvalues and Eigenvectors | 521 |

Solutions | 139 |

Projections and Projection Matrices | 161 |

Determinants | 179 |

Linear Transformations | 589 |

621 | |

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### Common terms and phrases

according arbitrary basis called Clearly column vector conclude condition consequence Consider consistent continuously differentiable Conversely Corollary corresponding decomposition deﬁned derivative determinant diagonal elements diagonal matrix dimensions eigenvalues eigenvectors elements equality equations equivalently essentially disjoint established example Exercise exists expression ﬁnd ﬁrst following theorem formula function Further hence idempotent implying inner product integers inverse least Lemma Let A represent light of Lemma linear space linear system linear transformation linearly independent Moreover multiplicity n n matrix n n symmetric n-dimensional nonnegative deﬁnite matrix nonnull nonsingular Note observing obtained orthogonal orthogonal matrix permutation polynomial Proof rank rank.A referred represent an n n representation respect result rows scalar Show Similarly solution square submatrix subspace Suppose symmetric matrix symmetric nonnegative deﬁnite unique upper triangular usual variables x0Ax