A theorem on matrices of 0’s and 1’s
Michael P. Drazin, Emilie V. Haynsworth · Pacific Journal of Mathematics · 1963
In this note we define two types of matrices, called "special" and "quasi-special", which we first discuss in their own rights; it turns out that the quasi-special matrices have a canonical representation (under permutational similarity) in terms of special matrices.We show how this fact can, essentially, be expressed in the language of graph theory, and we also use it to give a new proof of a theorem of Goldberg [1] on matrices with real roots.We shall be concerned, specifically, with the following properties of an n x n matrix A = (α«):DEFINITION 1.We call A special if a iό Φ 0 implies a H Φ 0.DEFINITION 2. Given any integer s with 1 3 g s ^ n, we call A s-special if, for every ordered set (i) = (i lf , i s ) of integers i r in the range 1 ^ i r rg n (r = 1, , s), the statement NΛΐ): a hh Φ 0, --, α fV _ lίf Φ 0 , a hh Φ 0 implies NΛΐ): a hh Φ 0, -, a isis _ x Φ 0 , a iχU Φ 0 .For example, every symmetric matrix is special (and the same is true of hermitian matrices over any ring with involution).Also, obviously, every special n x n matrix is s-special for each s = 3, , n, and it will be convenient to call any matrix with this latter property quasispecial.Thus every special matrix is quasi-special.The converse of this is easily seen to be false: e.g.is 3-special (since N^(ii, i 29 i 3 ) is always false), hence quasi-special, but this A is evidently not special.Nevertheless, every quasi-special matrix does have certain special matrices associated with it.More precisely, our main result is