The Enhanced Principal Rank Characteristic Sequence

Abstract

<p>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 <em>k</em> are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the <em>k</em>th entry is 0 or 1 according as none or at least one of its principal minors of order <em>k</em> 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.</p>