A matrix of permanents
If A is an \(n\times(n-2)\) matrix let \(\bar A\) be the \(n\times n\) matrix with zero diagonal and whose (i,j) element, for \(i\neq j\), is the permanent of the matrix obtained by deleting the \(i^{th}\) and \(j^{th}\) rows of A. If A is positive then \(\bar A\) is nonsingular and has exactly one positive eigenvalue. This fact is used in the proof of the van der Waerden conjecture [cf. \textit{J. H. Van Lint}, ibid. 39, 1-8 (1981; Zbl 0468.15005)]. The author proves that if A is nonnegative then \(\bar A\) is nonsingular if and only if A has no zero submatrix of n-1 lines. Using this he shows that if \((y^ t\bar Ax)^ 2=(y^ t\bar Ay)(x^ t\bar Ax),\) where x and y are column vectors with \(y\geq 0\), then one of the following is true: (i) \(x=\alpha y\) for some real \(\alpha\), (ii) at most one component of y is positive, (iii) A has a zero submatrix of n-1 lines.
scientific article; zbMATH DE number 3845758