Special matrices and their application in numerical mathematics. (Speciální matice a jejich použití v numerické matematice).

*(Czech)*Zbl 0531.65008
Teoretická Knižnice Inženýra. Praha: SNTL - Nakladatelství Technické Literatury. 266 p. Kcs 39.00 (1981).

In the book the properties of many important special matrices are studied: symmetric and Hermitian matrices, especially positive definite and positive semidefinite matrices, nonnegative matrices, regular and singular K- and \(K_ 0\)-matrices, diagonally dominant matrices, stable matrices, band matrices and large sparse matrices. The main attention is given to properties of special matrices which have practical numerical applications. For the investigation of these matrices the methods of the theory of general matrices and of the theory of oriented and unoriented graphs and bigraphs are used. The theoretical results about these matrices are used for an estimation of eigenvalue location of matrices, for an analysis of direct and iterative methods for the solution of system of linear equations, for matrix inversion and for the solution of the partial and the total problem of eigenvalue calculation. Many results in the book are original due to author; other results are gathered by the author from various journals.

The book consists of 15 chapters. Each chapter contains examples and exercises. Their solution is not given in the book. The most important results and theorems are proved. After the introduction the list of denotation used in the book is given. The 1st chapter defines basic concepts from matrix theory and gives the most important theorems. In the 2nd chapter symmetric and Hermitian matrices are studied. The Schur theorem and some other theorems about the splitting of these matrices are proved. Properties of positive definite and positive semidefinite matrices are described. Theorems about singular and polar decomposition of matrix are also proved. The 3rd chapter describes the connection between the finite graphs and matrices. Some basic definitions, properties and theorems on oriented and unoriented graphs are given. In the 4th chapter the class of nonnegative matrices is studied. The Perron- Frobenius theorem and its generalisation are proved. The properties of stochastic and doubly stochastic matrices are investigated. Matrices of class K and \(K_ 0\) and diagonally dominant matrices are studied in the 5th chapter. Tensor products and its properties are formulated in the 6th chapter. In this chapter also some compound matrices are studied arising as solution of some practical problems. Their spectral properties are investigated. The connection between polynomials and matrices are formulated in the 7th chapter. Attention is given also to stable matrices. Band matrices, especially tridiagonal matrices are investigated in the 8th chapter. An algorithm for approximate construction of row and column permutations of symmetric matrices is given which calculates the matrix with minimal band width. Vector and matrix norms and their generalisation are studied in the 9th chapter. The 10th chapter gives some important estimations of eigenvalue location of matrices.

Chapters 11-15 are devoted to numerical methods whereby for their construction and analysis theoretical results from the previous chapters are used. Direct methods for the solution of various systems of linear algebraical equations are studied in the 11th chapter. Properties of the pseudoinverse matrix are also mentioned. Iterative methods for the calculation of regular systems of linear equations are described and their convergence is studied in the 12th chapter (Jacobi, Gauss-Seidel and overrelaxation method). Methods of matrix inversion and some relations between various inverse matrices are given in the 13th chapter. The 14th chapter brings numerical methods for the calculation of eigenvalues (the power method and inverse iteration, Krylov’s, Householder’s and Jacobi’s method and the QR algorithm). The partial and the total problem of eigenvalue calculation are studied. The last 15th chapter is devoted to the numerical solution of system of linear equations with large sparse regular matrices. By graph theory the fill-in in elimination methods is investigated.

I think, that until now in the literature there is no such well understandable book devoted to special matrices and their applications. The book is written both for mathematicians and also for practical oriented users of linear algebra. I recommend to translate it from Czech to English as soon as possible.

The book consists of 15 chapters. Each chapter contains examples and exercises. Their solution is not given in the book. The most important results and theorems are proved. After the introduction the list of denotation used in the book is given. The 1st chapter defines basic concepts from matrix theory and gives the most important theorems. In the 2nd chapter symmetric and Hermitian matrices are studied. The Schur theorem and some other theorems about the splitting of these matrices are proved. Properties of positive definite and positive semidefinite matrices are described. Theorems about singular and polar decomposition of matrix are also proved. The 3rd chapter describes the connection between the finite graphs and matrices. Some basic definitions, properties and theorems on oriented and unoriented graphs are given. In the 4th chapter the class of nonnegative matrices is studied. The Perron- Frobenius theorem and its generalisation are proved. The properties of stochastic and doubly stochastic matrices are investigated. Matrices of class K and \(K_ 0\) and diagonally dominant matrices are studied in the 5th chapter. Tensor products and its properties are formulated in the 6th chapter. In this chapter also some compound matrices are studied arising as solution of some practical problems. Their spectral properties are investigated. The connection between polynomials and matrices are formulated in the 7th chapter. Attention is given also to stable matrices. Band matrices, especially tridiagonal matrices are investigated in the 8th chapter. An algorithm for approximate construction of row and column permutations of symmetric matrices is given which calculates the matrix with minimal band width. Vector and matrix norms and their generalisation are studied in the 9th chapter. The 10th chapter gives some important estimations of eigenvalue location of matrices.

Chapters 11-15 are devoted to numerical methods whereby for their construction and analysis theoretical results from the previous chapters are used. Direct methods for the solution of various systems of linear algebraical equations are studied in the 11th chapter. Properties of the pseudoinverse matrix are also mentioned. Iterative methods for the calculation of regular systems of linear equations are described and their convergence is studied in the 12th chapter (Jacobi, Gauss-Seidel and overrelaxation method). Methods of matrix inversion and some relations between various inverse matrices are given in the 13th chapter. The 14th chapter brings numerical methods for the calculation of eigenvalues (the power method and inverse iteration, Krylov’s, Householder’s and Jacobi’s method and the QR algorithm). The partial and the total problem of eigenvalue calculation are studied. The last 15th chapter is devoted to the numerical solution of system of linear equations with large sparse regular matrices. By graph theory the fill-in in elimination methods is investigated.

I think, that until now in the literature there is no such well understandable book devoted to special matrices and their applications. The book is written both for mathematicians and also for practical oriented users of linear algebra. I recommend to translate it from Czech to English as soon as possible.

Reviewer: J.Mikloŝko

##### MSC:

65Fxx | Numerical linear algebra |

15B57 | Hermitian, skew-Hermitian, and related matrices |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

15B51 | Stochastic matrices |

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

15A54 | Matrices over function rings in one or more variables |

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

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

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |