The permanents of k-tridiagonal toeplitz matrices

This thesis is based on the permanents of k-tridiagonal matrices with Toeplitz structure. In this study, both recursive and non-recursive combinatorial formulas for permanents of k-tridiagonal Toeplitz matrices, which have structured both the circulant and the band matrix, were tried to be obtained. This thesis mainly divided into six chapters. The first chapter covers the discussion of permanent function of the matrix family and its associated matrix structures. In the second chapter, a literature review on the subject is given. In the third chapter, the relationships between the permanents of a special kind of k-tridiagonal Toeplitz matrices (the main diagonal band with variable and the other bands include complex unit) and Chebyshev polynomials, which are important members of the orthogonal polynomials family, are presented. In the fourth chapter, the general k-tridiagonal Toeplitz matrices are studied and the recursive formulas obtained for these permanent are given. There is given also an algorithm based on these recursive formulas. The fifth chapter presents the combinational formulas obtained by using the recursive relations given in the previous chapter for the general k-tridiagonal Toeplitz matrices. In the last chapter, some interpretations have been made on the obtained results, and some of problems that arise out of this study are emphasized. The recursive relations appeared in this study are obtained by applying Laplace expansion on the permanents. Both the induction principle and combinatorial approximation are tested to prove those findings.