## The Theory of MatricesThis volume offers a concise overview of matrix algebra's many applications, discussing topics of extensive research and supplying proofs. Its contents include reviews of matrices, arrays, and determinants; the characteristic equation; associated integral matrices; equivalence, congruence, and similarity; composition of matrices; matric equations; functions of matrices; and matrices of infinite order. 1946 edition. |

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### Contents

Matrices Arrays and Determinants | 1 |

Total niatric algebra | 2 |

Diagonal and scalar matrices | 5 |

Determinants | 6 |

Properties of determinants | 8 |

Rank and nullity | 10 |

Identities among minors | 12 |

Reducibility | 14 |

Linear form moduls | 37 |

Ideals | 38 |

Equivalence | 40 |

Invariant factors and elementary divisors | 43 |

Factorization of a matrix | 44 |

29 Polynomial domains | 45 |

Equivalent pairs of matrices | 48 |

Automorphic transformations | 50 |

Arrays and determinants of higher dimension | 15 |

Matrices in noncommutative systems | 16 |

The characteristic equation | 17 |

15 Determination of the minimum equation | 20 |

Characteristic roots | 22 |

Conjugate sets | 24 |

Limits for the characteristic roots | 25 |

Characteristic roots of unitary matrices | 28 |

HI Associated Integral Matrices | 29 |

Construction of unimodular matrices | 31 |

Greatest common divisors | 35 |

Congruence | 51 |

33 Matrices with rational integral elements | 54 |

Matrices with elements in a field | 56 |

35 Matrices in an algebraically closed field | 60 |

Hermitian matrices | 62 |

Automorphs | 65 |

Similarity | 68 |

Composition of matrices | 81 |

Matric equations | 89 |

Functions of Matrices | 97 |

Matrices of infinite order | 104 |

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