## Matrices and Their Roots: A Textbook of Matrix AlgebraThis textbook addresses itself to two groups of students who need mathematics in an applied context: undergraduates starting at the beginning, and postgraduates who need reference-material, but who, not being mathematics specialists, nevertheless are not best served by an ordinary mathematics textbook, which will generally be at a higher level of abstraction. It gives full proofs throughout, and is illustrated with a large number of numerical examples, reinforcing the student's grasp of the topics covered by exercises and corresponding answersheets, and by the corresponding tutorial program ILLUSTRATE. The program ‘Illustrate’ will run on any IBM compatible micro-computer. The relevant areas of application are economics, econometrics, mathematical programming and engineering. |

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### Contents

Matrix Notation | 11 |

Some Structured Matrices and Operators | 38 |

BlockEquations and Inversion | 51 |

Determinants and Rank | 84 |

Definiteness and Symmetry | 142 |

Latent Roots and Characteristic Vectors | 165 |

More about Roots and Vectors | 208 |

Symmetric Eigenvalue Problems | 279 |

Geometrical Interpretations | 326 |

NonNegative Square Matrices | 384 |

431 | |

Indexes | 437 |

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### Common terms and phrases

absolute value algebra Answersheet applies assumed block-column blockdiagonal blockpermutoid calculation tableau chapter characteristic equation characteristic vectors coefficients column vectors complex numbers complex roots conform convex combination decomposable determinant develop diagonal blocks diagonal elements diagonal matrix differentiation elimination example first-order formula full rank idempotent illustrated implies indecomposable independent vectors inversion step invertible matrix Jordan block leading row lefthand multiplication non-zero elements normalized null matrix null vector obtain origin orthogonal outer product partitioning permits permutation operator permutoid positive definite positive pivots pre-multiplication previous section principal minors proof by recursive quadratic form re-ordering real root recursive induction recursive product refer relation repeated root requirement righthand side rotation operator row-operations row-vector satisfying semi-positive square matrix Subcase subtraction symmetric matrix symmetric product expression term theorem is true third row transformation transpose triangular unity elements variables vectors associated x'Ax Xi X2 Xl X2 X3 zero root