## Matrices: Theory and ApplicationsIn this book, Denis Serre begins by providing a clean and concise introduction to the basic theory of matrices. He then goes on to give many interesting applications of matrices to different aspects of mathematics and also other areas of science and engineering. The book mixes together algebra, analysis, complexity theory and numerical analysis. As such, this book will provide many scientists, not just mathematicians, with a useful and reliable reference. It is intended for advanced undergraduate and graduate students with either applied or theoretical goals. This book is based on a course given by the author at the Ecole Normale Superieure de Lyon. Denis Serre is Professor of Mathematics at Ecole Normale Superieure de Lyon and a former member of the Institut Universaire de France. He is a member of numerous editorial boards and the author of Systems of Conservation Laws (Cambridge University Press 2000). The present book is a translation of the original French edition, Les Matrices: Theorie et Pratique, published by Dunod (2001). |

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### Contents

III | 1 |

IV | 8 |

V | 13 |

VI | 15 |

VII | 19 |

VIII | 21 |

IX | 23 |

X | 24 |

XXXVI | 111 |

XXXVII | 114 |

XXXVIII | 116 |

XXXIX | 120 |

XL | 122 |

XLI | 123 |

XLII | 127 |

XLIII | 128 |

XI | 28 |

XII | 29 |

XIII | 30 |

XIV | 31 |

XV | 40 |

XVI | 43 |

XVII | 45 |

XVIII | 47 |

XIX | 51 |

XX | 55 |

XXI | 61 |

XXII | 66 |

XXIII | 67 |

XXIV | 70 |

XXV | 71 |

XXVI | 73 |

XXVII | 80 |

XXVIII | 81 |

XXIX | 82 |

XXX | 85 |

XXXI | 87 |

XXXII | 91 |

XXXIII | 97 |

XXXIV | 101 |

XXXV | 104 |

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### Common terms and phrases

algebraic algorithm associated assume basis bistochastic block-diagonal blockwise canonical characteristic polynomial column companion matrix complex computation conjugate convex Deduce deﬁne detM diagonal blocks diagonal entries diagonal matrix diagonalizable eigenvalues eigenvector element equal equivalent Euclidean exercise exists ﬁeld ﬁnite ﬁrst formula given Hence Hermitian matrix Hessenberg matrix homeomorphic implies induced norm induction inequality invertible irreducible isomorphism Jacobi method Lemma Let us define Let us denote linear subspace LU factorization matrix norm method converges minimal polynomial Mn(CC Mn(IR Mn(K multiplicity nonnegative nonzero obtain orthogonal permutation permutation matrix polar decomposition positive deﬁnite principal ideal domain Proof Let Proposition QR method real numbers real symmetric roots satisﬁes scalar sequence Show similarity invariants spectrum square matrix subgroup symmetric matrix Theorem triangular matrix tridiagonal unique unitary matrices upper triangular vector space zero