Abstract
It is known that the so-called van der Waerden's conjecture, regarding doubly stochastic matrices, was solved in full generality in 1980 and 1981, respectively. In this paper, we deal with equations regarding stochastic matrices generated by double stochastic matrices. Let the quantities t k ( A), ( k = 0, 1, …, n) be defined by (1.1), where A is an n × n doubly stochastic matrix. Moreover, let the system of operators C (Definition 1.1) be given. The results of the paper are the following. If the upper permanent and the lower permanent of two stochastic matrices are equal, then at least one of the factors is equal to A 0, where A 0 is the matrix with entries 1 n (Theorem 1.1). Theorems 1.2–1.4 deal with the means of permanents. An inequality (Lemma 1.2) and a consequence of it (Lemma 1.3) are used in the proofs. These results were obtained when the author was a student in 1932. The paper underlines the importance of the concomitant (Definition 1.2) of a matrix.
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