Some results on Hadamard closure and variation diminishing properties of totally nonnegative matrices

It has long been known that totally nonnegative (or totally positive matrices) are closed under normal matrix multiplication. A 2001 paper by Crans, Fallat and Johnson titled the “Hadamard Core of Totally Nonnegative Matrices\u27\u27, described completely the subset of up to 3 × n and n × 3 TN matrices for which a modified closure property for Hadamard multiplication holds. In the paper, they also conjectured a set of test matrices that might work in the 4 × 4 case. This dissertation begins with a result in this area by showing that the conjecture is true for 4 × 4 TN matrices with a zero on the tridiagonal part. It also expands the symmetry used in the 3 × 3 case to the more general n × n case which may be helpful for higher order proofs. In addition, the relationship of TN matrices with the variation diminishing property is explored. For a strictly nonzero vector x in Rn, successive coordinates either have positive or negative product. A sign change in the vector is recorded whenever this product is negative. However, if the vector were allowed to have entries equal to 0, then there are a maximum and minimum number of sign changes of x, denoted V+(x) and V_(x), respectively, based on whether the zero entries were replaced with positive or negative entries. When a matrix A is such that V+(Ax) ≤ V_(x) for all x in a set S, then A is said to be variation diminishing on S. This thesis presents results on the variation diminishing properties of TP and invertible TN matrices for all strictly nonzero vectors x using the factorization of such matrices into elementary bidiagonal TN matrices