The Perron-Frobenius Theory And Its Deduction Based On The Wielandt Method

Algebra representation theory is a new algebraic branch arising in 1970s whose researches mainly focuses on rings and algebraic structures. Within thirty years, this theory developed rapidly and becoming perfect.Perron firstly discovered the spectrum properties about square matrix in 1907. After that, during year 1908 to 1912, these results have been expanded to cases about nonnegative matrices, by Frobenius, especially concerned with the nonnegative irreducible matrix. In year 1973 to 1975, there are also satiable results derived from the researches on cases of reducible matrix.The study of the spectrum properties is both theoretically and practically valuable. Its use in spectrum analysis of all kinds of matrices, especially in the theory of Markov, answer of equations, the general theory about the numerical solution of partial differential equations, is always a popular question concerned by scientists.Based on such results and their applications, the properties and structure of spectrum were studied in this paper. This paper tries to introduce the Perron-Frobenius theory on nonnegative irreducible matrix (including square matrix) and adopts Wielandt's method to deduce it. (There are many ways to derive this theory, see reference [5] and [6].) By doing so, the classical results of Perron-Frobenius theory on common nonnegative matrix and their generalizations are derived. Some of these corollaries can provide boundary estimation on spectrum's radius and its structural analysis of nonnegative irreducible matrix. These results are important theoretically, especially within the realm of matrix iteration analysis.