We show that an n × n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions (except the first and last) completely between the subdiagonal and superdiagonal part, then this matrix can be factorized into a product of n-1 matrices, each consisting of a 2 × 2 principal submatrix in two consecutive rows (and columns) in all possible of the n-1 positions, and completed by ones along the diagonal. The converse is also true. It is shown that the spectrum does not depend on the order of the factors. © 2004 Elsevier Inc. All rights reserved.
Fiedler, M. (2004). Complementary basic matrices. Linear Algebra and Its Applications, 384(1-3 SUPPL.), 199–206. https://doi.org/10.1016/j.laa.2004.01.014