##
**The theory of matrices. 2nd ed., with applications.**
*(English)*
Zbl 0558.15001

Computer Science and Applied Mathematics. Orlando etc.: Academic Press (Harcourt Brace Jovanovich, Publishers). XV, 570 p. $ 59.00 (1985).

This is the second edition of the well known book. The first edition was published by the first author (1969; Zbl 0186.053)]. The second edition contains not only much new material but also a new presentation of the old material. The basic objectives of the authors are: 1) to give a self- contained presentation of the theory of matrices and linear finite- dimensional spaces, 2) to present applications of the theory, especially to stability problems in ordinary differential and difference equations, 3) to give the material necessary for the deep study of the computational methods of linear algebra. The first seven chapters contain the expanded basic course in the theory of matrices and linear algebra including the Jordan canonical form and the theory of elementary divisors, the Smith canonical form, and applications to differential and difference equations. Chapters 8-11 and 15 (Variational methods, Functions of matrices, Norms and bounds for eigenvalues, Perturbation theory, Nonnegative matrices) contain much material which is used in many applications in physics, economics, engineering, etc. Chapters 12-14 (Linear matrix equations and generalized inverses, Stability problems, Matrix polynomials) contain new material some of which, especially in chapter 14, is of recent origin [see I. Gohberg, the first author and L. Rodman, Matrix polynomials (1982; Zbl 0482.15001)]. This well written book contains many exercises and can be used by undergraduate and graduate students in a number of courses. Specialists will be glad to have this book for references.

Reviewer: A.G.Ramm

### MSC:

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

34Dxx | Stability theory for ordinary differential equations |

39A11 | Stability of difference equations (MSC2000) |

93Dxx | Stability of control systems |

65Fxx | Numerical linear algebra |

### Keywords:

monograph; theory of matrices; theory of elementary divisors; Variational methods; Functions of matrices; Norms and bounds for eigenvalues; Perturbation theory; Nonnegative matrices; Linear matrix equations; generalized inverses; Stability problems, Matrix polynomials; exercises### Online Encyclopedia of Integer Sequences:

Triangle T(n,k) in which the n-th row encodes the inverse of a 3n+1 X 3n+1 Jacobi matrix, with 1’s on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n+1).Triangle T(n,k) in which the n-th row encodes the inverse of a 3n X 3n Jacobi matrix, with 1’s on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n).