The Rank-Preserving Transversality Property

In a paper by M. Arav, F.J. Hall, H. van der Holst, Z. Li, A. Mathivanan, J. Pan, H. Xu, Z. Yang, “Advances on similarity via transversal intersection of manifolds,” Linear Algebra Appl. 688 (2024), 1-20, the authors introduce the similarity-transversality property (STP). The paper defines the STP of n × n matrix A as the full row rank property of the Jacobian matrix of the entries of AX − XA at the zero entry positions of A with respect to the nondiagonal entries of X, where X = [aij ] is a generic matrix of order n whose entries are independent variables. These authors show a number of properties of STP matrices and classes of zero-nonzero patterns and sign patterns that require or allow the STP. This work is inspired by the previously mentioned paper as well as “A matching-minor monotone parameter for bipartite graphs,” Linear Algebra Appl. 680 (2024) 254-273, by M. Arav, L. Deaett, H.T. Hall, H. van der Holst, and D. Young, where the authors use the asymmetric strong Arnold property (ASAP). When restricted to simple bipartite graphs, the ASAP is equivalent to the property introduced in this dissertation: the rank-preserving transversality property (RPTP). Let B be a m × n real matrix. Then if the manifolds eM B = {H−1BG : G,H are nonsingular} and Q(sgn(B)) intersect transversally at B, that is, the tangent spaces of eMB and Q(sgn(B)) at B sum to Rm×n, we say B has the RPTP. This dissertation establishes many important properties of matrices with the RPTP as well as several sign pattern classes that require the RPTP. For example, it is shown that the RPTP matrices are closed under permutation equivalence, diagonal equivalence, and transpose. Futher, it is shown that a block upper triangular matrix has the RPTP if and only if each diagonal block has the RPTP and at most one diagonal block is singular. A technique for partitioning a matrix to more easily check if it has the RPTP is presented. Additionally, two related properties, the row-equivalence transversality property (RETP) and the columnequivalence transversality property (CETP), are introduced and investigated.