Matrices. Theory and applications. Transl. from the French.

*(English)*Zbl 1011.15001
Graduate Texts in Mathematics. 216. New York, NY: Springer. xv, 202 p. (2002).

This text offers an advanced course in matrix theory aimed at a student with a good background in analysis, providing a selection of advanced topics in matrices over the real or complex fields without attempting to be encyclopedic.

The first three chapters review basic notation and theorems. The following chapters deal with: norms, convergence and Gerschgorin domains; the Perron-Frobenius theorem for nonnegative matrices, and Birkhoff’s characterization of doubly stochastic matrices; invariant factors of matrices and the Jordan form; polar decomposition, the exponential of a matrix, and topological properties of the classical linear groups; matrix factorizations (LU-, QR-factorisation and singular value decomposition) and the Moore-Penrose generalized inverse; iterative methods for solving systems of linear equations; approximation of eigenvalues via the QR-method, the Jacobi method and the power method.

The last two chapters which deal with problems in numerical analysis look at the subject from the point of view of an analyst without any substantial discussion of questions which arise when calculations are carried out to limited precision.

The book contains a large number of exercises, many interesting and challenging, including extensions of results discussed in the main text and alternative proofs of some theorems.

The first three chapters review basic notation and theorems. The following chapters deal with: norms, convergence and Gerschgorin domains; the Perron-Frobenius theorem for nonnegative matrices, and Birkhoff’s characterization of doubly stochastic matrices; invariant factors of matrices and the Jordan form; polar decomposition, the exponential of a matrix, and topological properties of the classical linear groups; matrix factorizations (LU-, QR-factorisation and singular value decomposition) and the Moore-Penrose generalized inverse; iterative methods for solving systems of linear equations; approximation of eigenvalues via the QR-method, the Jacobi method and the power method.

The last two chapters which deal with problems in numerical analysis look at the subject from the point of view of an analyst without any substantial discussion of questions which arise when calculations are carried out to limited precision.

The book contains a large number of exercises, many interesting and challenging, including extensions of results discussed in the main text and alternative proofs of some theorems.

Reviewer: J.D.Dixon (Ottawa)

##### MSC:

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |

15A42 | Inequalities involving eigenvalues and eigenvectors |

15B48 | Positive matrices and their generalizations; cones of matrices |

15B51 | Stochastic matrices |

15A21 | Canonical forms, reductions, classification |

20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |

15A23 | Factorization of matrices |

65F05 | Direct numerical methods for linear systems and matrix inversion |

65F10 | Iterative numerical methods for linear systems |

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

15A09 | Theory of matrix inversion and generalized inverses |

65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |