The Enhanced Principal Rank Characteristic Sequence

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2016-06-01
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Butler, Steve
Catral, Minerva
Fallat, Shaun
Hall, H. Tracy
Hogben, Leslie
van den Driessche, P.
Young, Michael
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Electrical and Computer EngineeringMathematics
Abstract

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n×n matrix is a sequence ℓ12⋯ℓn where ℓk 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.

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This is a manuscript of an article published as Butler, Steve, Minerva Catral, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Pauline van den Driessche, and Michael Young. "The enhanced principal rank characteristic sequence." Linear Algebra and its Applications 498 (2016): 181-200. DOI: 10.1016/j.laa.2015.03.023. Posted with permission.

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Thu Jan 01 00:00:00 UTC 2015
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