Continued Fractions And Casorati Matrices

In this paper the relations between the periodicity of continued fractions, Casorati matrices and Diophantine equation are established by using the matrix formulation of continued fraction theory.First of all, the expression of Casorati matrix in continued fraction theory and its important properties are obtained from the recurrence relations of the numerators and denominators p nand qn of convergents. Then, by using the nonnegativity and irreducibility of Casorati matrix and Perron-Frobenius theorem, the relations between its positive eigenvetor which corresponds to the largest eigenvalue and the periodicity of continued fractions are researched. At last, the similarity among the product matrices which start from different partial quotients in a periodic section is studied. In this article we provide a new method to solve the smallest positive solution of the linear and nonlinear Diophantine equations, especially Pell's equation, by using Casorati matrix theory, which differs form the Euler's method and general continued method.The content of the article is common, but the research program we provide is very original. An intimate relation between the solutions of Pell's equation and the periodicity of the partial quotient sequence of continued fractions is clearly shown by quoting matrix theory. In a word, we mainly try to find out a new method and idea to solve common problems regarding of continued fractions.In order to emphasize the evidence and practicality of the theory, we take the solution of a simple linear Diophantine equation by using finite continued fraction theory as an example in the second part, certainly, which also can be solved by matrix theory. In the third part, we confirm some important properties of matrices corresponding to pure recurring continued fractions. In the last part, we continue confirming the correct of the matrix method of solution of Pell's equations by two simple examples.