Orthogonal bases that lead to symmetric nonnegative matrices

In a paper dating back to 1983, Soules constructs from a positive vector x an orthogonal matrix R which has the property that for any nonnegative diagonal matrix Λ with nonincreasing diagonal entries, the matrix RΛRT has all its entries nonnegative. Independently, Fiedler in 1988 showed that any symmetric irreducible nonsingular matrix whose powers are all M-matrices (and hence an MMA-matrix in the language of Friedland, Hershkowitz, and Schneider) must have an orthogonal matrix of eigenvectors R¯ which has similar properties to those of R. Here, for a given positive n-vector x, we investigate the structure of all orthogonal matrices R for which, for any nonnegative diagonal matrix Λ as above, the matrices RΛRT are nonnegative. Up to a permutation of its columns, each such R corresponds to a binary tree whose vertices are subsets of the set 1, 2, …, n with the property that each vertex has either no successor or exactly two disjoint successors. For such orthogonal matrices R and such nonsingular diagonal matrices Λ, we show that the set of matrices of the form RΛRT and the set of inverse MMA-matrices (i.e. matrices whose inverses are MMA-matrices) coincide. Using this result, we establish a relation between strictly ultrametric matrices and inverse MMA-matrices. Finally, we show that the QR factorization of RΛRT, for certain such Rʹs, has a special sign pattern.