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Full Description
A circulant matrix is one in which a basic row of numbers is repeated again and again, but with a shift in position. Such matrices have connection to problems in physics, signal and image processing, probability, statistics, numerical analysis, algebraic coding theory, and many other areas. At the same time, the theory of circulants is easy, relative to the general theory of matrices. Practically every matrix-theoretic question for circulants may be resolved in closed form. Consequently, circulant matrices constitute a nontrivial but simple set of objects that the reader may use to practice, and ultimately deepen, a knowledge of matrix theory. They can also be viewed as special instances of structured or patterned matrices. This book serves as a general reference on circulants, as well as provides alternate or supplemental material for intermediate courses on matrix theory. There is some general discussion of matrices: block matrices, Kronecker products, decomposition theorems, generalised inverses. These topics were chosen because of their application to circulants and because they are not always found in books on linear algebra. More than 200 problems of varying difficulty are included.
Contents
An Introductory Geometrical Application: 1.1 Nested triangles; 1.2 The transformation $\sigma$; 1.3 The transformation $\sigma$, iterated with different values of $s$; 1.4 Nested polygons Introductory Matrix Material: 2.1 Block operations; 2.2 Direct sums; 2.3 Kronecker product; 2.4 Permutation matrices; 2.5 The Fourier matrix; 2.6 Hadamard matrices; 2.7 Trace; 2.8 Generalized inverse; 2.9 Normal matrices, quadratic forms, and field of values Circulant Matrices: 3.1 Introductory properties; 3.2 Diagonalization of circulants; 3.3 Multiplication and inversion of circulants; 3.4 Additional properties of circulants; 3.5 Circulant transforms; 3.6 Convergence questions Some Geometric Applications of Circulants: 4.1 Circulant quadratic forms arising in geometry; 4.2 The isoperimetric inequality for isosceles polygons; 4.3 Quadratic forms under side conditions; 4.4 Nested $n$-gons; 4.5 Smoothing and variation reduction; 4.6 Applications to elementary plane geometry: $n$-gons and $K_r$-grams; 4.7 The special case: $\text{circ}(s, t, 0, 0, \dots, 0)$; 4.8 Elementary geometry and the Moore-Penrose inverse Generalizations of Circulants: $g$-Circulants and Block Circulants: 5.1 $g$-circulants; 5.2 $0$-circulants; 5.3 PD-matrices; 5.4 An equivalence relation on $\{1, 2, \dots, n\}$; 5.5 Jordanization of $g$-circulants; 5.6 Block circulants; 5.7 Matrices with circulant blocks; 5.8 Block circulants with circulant blocks; 5.9 Further generalizations Centralizers and Circulants; 6.1 The leitmotiv; 6.2 Systems of linear matrix equations. The centralizer; 6.3 $\div$ algebras; 6.4 Some classes $Z(P_{\sigma}, P_{\tau})$; 6.5 Circulants and their generalizations; 6.6 The centralizer of $J$; magic squares; 6.7 Kronecker products of $I, \pi$, and $J$; 6.8 Best approximation by elements of centralizers Appendix Bibliography Index of authors Index of subjects
