Iohvidov I.S. Henkel and Toeplitz Matrices and Forms. Algebraic Theory

Translated by G.Philip, A. Thijsse. — Boston; Basel; Stutgart: Birkhauser, 1982. — 231 p.This book is valuable introduction to the theory of finite Henkel and Toeplitz matrices. These matrices are characterized by the property that is one of them the entries of the matrices depend only on the sum of indices, and in the other only on the differences of the indices.
The book is dedicated in general to the algebraic aspect of the theory and the main attention is given to problems of: extensions, computations of ranks, signature, and inversion. The author has succeeded in presenting these problems in a unified way, combining basic material with results.
Henkel and Toeplitz matrices have a long history and have given rise to important recent applications (numerical analysis, system theory, and others).
The book is self-contained and only a knowledge of a standard course in linear algebra is required of the reader. The book is nicely written and contains a system of well chosen exercises. The book can be used as a text book for graduate and senior undergraduate students.
(From editorial introduction by I. Gohberg).Some information from the general theory of martices and forms
The reciprocal matrices and its minors
The Sylvester identities for bordered minors
Evaluation of certain determinants.
Matrices and linear operators Spectrue
Hermitian and quadratic forms. Law of inertia. Signature
Truncated forms
The Sylvester formula and the representation of a Hermitian form as a sum of squares by the method of Jacobi
The signature rule of Jacobi and its generalizationHenkel matrices and forms
Henkel matrices. Singular extensions
The (r,k)- charachteristic of a Henkel matrix
Theorems on the rank
Henkel formsToeplitz matrices and forms
Toeplitz matrices. Singular extensions
The (r,k,l)- charachteristic of Toeplitz matrices
Theorems on the rank
Hermitian Toeplitz formsTransformations of the Toeplitz and Henkel matrices and forms
Mutual transformation of Toeplitz and Henkel matrices. Recalculation of the charactristics
Inversion of Toeplitz and Henkel matrices
Mutual transformation of Toeplitz and Henkel formsAppendices.
The theorem of Borhardt-Jacobi and of Herglotz-M. Krein on the roots of real and Hermitian-symmetric polynomials. Note to App. 1
The functionals S and C and some of their applications. Notes to App. 2