An elementary proof that every singular matrix is a product of idempotent matrices

• Autor(a) principal: Araújo, João
• Data de Publicação: 2005
• Outros Autores: Mitchell, James D.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.2/3784

In this note we give an elementary proof of a theorem first proved by J. A. Erdos [3]. This theorem, which is the main result of [3], states that every noninvertible n ⇥ n matrix is a finite product of matrices M with the property that M2 = M. (These are known as idempotent matrices. Noninvertible matrices are also called singular matrices.) An alternative formulation of this result reads: every noninvertible linear mapping of a finite dimensional vector space is a finite product of idempotent linear mappings ↵, linear mappings that satisfy ↵2 = ↵. This result was motivated by a result of J. M. Howie asserting that each selfmapping ↵ of a nonempty finite set X with image size at most |X|−1 (which occurs precisely when ↵ is noninvertible) is a product of idempotent mappings. We shall see that Erdos’s theorem is a consequence of Howie’s result. Together the papers [3] and [4] are cited in over one hundred articles, dealing with subjects including universal algebra, ring theory, topology, and combinatorics. Since its publication, various proofs of the result in [3] have appeared. For example, a semigroup theoretic proof appears in [1, pp. 121-131] and linear operator theory is used to prove the theorem in [2]. Here we give a new proof using a basic combinatorial argument. Unlike the previous proofs our argument involves only elementary results from linear algebra and one basic result concerning permutations. On the way to proving the main result of this note we provide a short proof of Howie’s result. Throughout this paper X signifies an arbitrary nonempty finite set. If ↵ : A ! X, where A is a subset of X, then A is the domain of ↵; we denote this set by dom(↵). Naturally, the set ↵(A) is called the image of ↵ and is denoted by im(↵). Recall that a mapping ↵ is injective (or one-to-one) if ↵(x) 6= ↵(y) for all x and y in dom(↵) with x 6= y. Let TX denote the set of all mappings from X to X with domain X. We note that this set is closed under composition of mappings and that this composition is associative. We now define one of the most important notions we require in the proofs in this note. For a mapping ↵ : dom(↵) ! X we say that ↵ is a restriction of an element " of TX if " and ↵ agree on the domain of ↵. In other words, "(x) = ↵(x) for all x in dom(↵). For x and y in X we denote the transposition that fixes every point of X other than x or y and that maps x to y, and vice versa, by (x y).