The enhanced principal rank characteristic sequence

Steve Butler, Iowa State University
Minerva Catral, Xavier University, Cincinnati, OH
Shaun M. Fallat, University of Regina, Regina, SK
H. Tracy Hall, Brigham Young University
Leslie Hogben, Iowa State University
P. van den Driessche, University of Victoria, Victoria, BC
Michael Young, Iowa State University

This is a manuscript of an article from Linear Algebra and its Applications 498 (2016):181, doi:10.1016/j.laa.2015.03.023. Posted with permission.

Abstract

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n×n matrix is a sequence ℓ₁ℓ₂⋯ℓ_n where ℓ_k is A, S, or N according as all, some, or none of its principal minors of order k are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the kth entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero). Various techniques including the Schur complement are introduced to establish that certain subsequences such as NAN are forbidden in epr-sequences over fields of characteristic not two. Using probabilistic methods over fields of characteristic zero, it is shown that any sequence of As and Ss ending in A is attainable, and any sequence of As and Ss followed by one or more Ns is attainable; additional families of attainable epr-sequences are constructed explicitly by other methods. For real symmetric matrices of orders 2, 3, 4, and 5, all attainable epr-sequences are listed with justifications.