Mathematics, Electrical and Computer Engineering
Journal or Book Title
Linear Algebra and its Applications
The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n×n matrix is a sequence ℓ1ℓ2⋯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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Butler, Steve; Catral, Minerva; Fallat, Shaun M.; Hall, H. Tracy; Hogben, Leslie; van den Driessche, P.; and Young, Michael, "The Enhanced Principal Rank Characteristic Sequence" (2016). Mathematics Publications. 52.