The Concepts of Irreducibility and Full Indecomposability of a Matrix in the Works of Frobenius, K6nig and Markov

Frobenius 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-Konig 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. K&rig 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 Markov 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. 0. Introduction and Motivution This article combines detailed examination of parts of classical and influential papers by G. F. Frobenius (1849-1917), A. A. Markov (185&1922), and D. Konig (188441944) (“mathematical history”‘)t with a new result and proof (“research”). Such a combination is appropriate in this instance, since our Theorem (7.1) clarifies some results in these papers and was conjectured after studying them. In Part I of this article we discuss the following topics listed in (0.1) through (0.4): (0.1) There is a theorem of Frobenius which characterizes irreducible matrices and fully indecomposable matrices; see (1.1) and (1.2) for definitions. This theorem is stated in identical words in Frobenius [1912] and *This research was supported in part by the National Science Foundation under grant MPS 73-08618 A02. +Superscripts refer to notes collected in Sec. 9. LINEAR ALGEBRA AND ITS APPLICATIONS 18, 139-162 (197T) 0 Elsevier North-Holland, Inc., 1977 139