The concepts of irreducibility and full indecomposability of a matrix in the works of Frobenius, König and Markov

Abstract

AbstractFrobenius published two proofs of a theorem which characterizes irreducible and fully indecomposable matrices in an algebraic manner. It is shown that the second proof, which depends on the Frobenius-König theorem, yields a stronger form of the result than the first. Some curious features in Frobenius's last paper are examined; these include his criticisms of a result due to D. König and the latter's application of graph theory to matrices. A condition on matrices formulated by Markov is examined in detail to show that it may coincide with Frobenius's concept of irreducibility, and several theorems on stochastic matrices of Perron-Frobenius type proved by Marcov are exhibited. In a research part of the paper, a theorem is proved which characterizes irreducible matrices and which contains Frobenius's theorem and was motivated by Markov's condition