The enhanced principal rank characteristic sequence

Date
2015
Authors
Butler, Steve
Catral, Minnie
Fallat, Shaun
Hall, Tracy
Hogben, Leslie
van den Driessche, Pauline
Young, michael
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Abstract

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n ×n matrix is a sequence from 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 ofAs andSs ending inAis attainable, and any sequence ofAs andSs followed by one or moreNs 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.

Description
Keywords
principal rank, symmetric matrices
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