I have come across a class of matrices with some interesting properties. I feel that they must be known but I have not been able to find anything written about them. This is probably just because I do not know the right place to look. I will describe these matrices here and I hope that somebody will be able to point out a source where I can find more information about them. Consider an matrix with elements having the following properties. The elements with (call them ) are negative. The elements with (call them ) are positive. All other elements are zero. The determinant of a matrix of this type is . Notice that the two terms in this sum always have opposite signs. A property of these matrices which I found surprising is that is a positive matrix, i.e. all its entries are positive. In proving this it is useful to note that the definition of the class is invariant under cyclic permutation of the indices. Therefore it is enough to show that the entries in the first row of are non-zero. Removing the first row and the first column from leaves a matrix belonging to the class originally considered. Removing the first row and a column other than the first from leaves a matrix where is alone in its column. Thus the determinant can be expanded about that element. The result is that we are left to compute the determinant of an matrix which is block diagonal with the first diagonal block belonging to the class originally considered and the second diagonal block being the transpose of a matrix of that class. With these remarks it is then easy to compute the determinant of the matrix resulting in each of these cases. In more detail and for .
Knowing the positivity of means that it is possible to apply the Perron-Frobenius theorem to this matrix. In the case that has the same sign as it follows that has an eigenvector all of whose entries are positive. The corresponding eigenvalue is positive and larger in magnitude than any other eigenvalue of . This vector is also an eigenvalue of with a positive eigenvalue. Looking at the characteristic polynomial it is easy to see that if the matrix has exactly one positive eigenvalue and that none of its eigenvalues is zero.