Generalized Circulant Matrices And Its Applications

Circulant matrices, which are an important component of the matrix theory, havebecome one of the most important and active research fields of applied mathematicsincreasingly. Because of many good properties and structures of circulant matrices, it isnecessary to generalize and discuss their special properties, special structures, all kindsof polynomial representations, diagonalization, spectral decomposition, nonsingularity,eigenvalues, characteristic polynomial and fast algorithms for computing minimalpolynomial, inverse, self-reflective g-inverse, group inverse and Moore-Penrose inverse, and so on.Detail as follows: 1. Three sufficient conditions and a sufficient and necessary condition for determining thenonsingularity of scaled factor circulant matrix are presented. By means of the fast algorithm forcomputing polynomials and Euclid algorithm, fast algorithms for calculating the inverse,self-reflective g-inverse, group inverse and Moore-Penrose inverse of scaled factor circulant matrixare proposed, respectively. When scaled factor circulant matrix is nonsingular, an interpolationalgorithm for finding the inverse of such matrix is given by using the interpolation method and itsspecial properties. 2. The definitions of the-first-and-last-sum (FLS) r -circulant matrix andthe-first-and-last-sum (FLS) r -retrocirculant matrix are proposed. We discuss theirbasic properties, inverse, spectral inverse and group inverse. The relationship between therank of FLS r -circulant matrix and the eigenvalues of basic one is discussed.Simultaneously, a sufficient and necessary condition for determining the nonsingularity ofFLSr -circulant matrix is presented. By means of the fast algorithm for computing polynomials,fast algorithms for calculating the inverse, self-reflective g-inverse and group inverse ofFLS r -circulant matrix are proposed. When FLS r -circulant (retrocirculant) matrix isnonsingular, an algorithm for finding the inverse of such matrix is given by using the Euclid method,and it is extended to compute the group inverse of singular FLS r -circulant matrix.Generalizing above definitions, FLSR -factor block circulant matrix and FLSR -factorblock retrocirculant matrix are introduced. The basic properties are discussed and asufficient and necessary condition for determining the nonsingularity of FLSR -factor blockcirculant matrix is presented. Finally, fast algorithms for finding the inverses of FLSR -factorblock circulant matrix and FLSR -factor block retrocirculant matrix are obtained byusing the right largest common factor of the matrix polynomial. 3. Fast algorithms for solving FLSr -circulant (retrocirculant) linear systems arepresented. The unique solution is obtained when FLSr -circulant (retrocirculant) matrix is nonsingular, and the special solution and general solution, singular. Fast algorithmsfor finding the unique solution of FLS R -factor block circulant (retrocirculant) linearsystems are given. Sufficient and necessary condition of the existence of the unique solutionand fast algorithms for finding the unique solution of the inverse problem of AX = bin the class of the FLS r -circulant (retrocirculant) matrices and FLS R -factor blockcirculant (retrocirculant) matrices are presented. 4. The level-m scaled factor circulant matrix over any field is introduced. Theirbasic properties, diagonalization, explicit representation and spectral decomposition arediscussed over complex field. Three sufficient conditions for determining the nonsingularity andan algorithm for finding the inverse of the level-m scaled factor circulant matrix are presented.It is proved that the ring consisting of all level-m scaled factor circulant matrix over a field isisomorphic to a factor ring of a polynomial ring in multivariable over the same field.Algorithms for finding the minimal polynomial, inverse, the c...