1. The signed enhanced principal rank characteristic sequence.
- Author
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Martínez-Rivera, Xavier
- Subjects
- *
MATHEMATICAL sequences , *HERMITIAN structures , *SYMMETRIC matrices , *MATHEMATICAL analysis , *RANKING - Abstract
The signed enhanced principal rank characteristic sequence (sepr-sequence) of an
Hermitian matrix is the sequence , where is either , , , , , , or , based on the following criteria: if B has both a positive and a negative order-k principal minor, and each order-k principal minor is nonzero.(respectively, ) if each order- k principal minor is positive (respectively, negative).if each order- k principal minor is zero.if B has each a positive, a negative and a zero order-k principal minor.(respectively, ) if B has both a zero and a nonzero order-k principal minor, and each nonzero order-k principal minor is positive (respectively, negative). Such sequences provide more information than theepr-sequence in the literature, where the k th term is either, , or based on whether all, none, or some (but not all) of the order- k principal minors of the matrix are nonzero. Various sepr-sequences are shown to be unattainable by Hermitian matrices. In particular, by applying Muir’s law of extensible minors, it is shown that subsequences such asand are prohibited in the sepr-sequence of a Hermitian matrix. For Hermitian matrices of orders , all attainable sepr-sequences are classified. For real symmetric matrices, a complete characterization of the attainable sepr-sequences whose underlying epr-sequence contains as a non-terminal subsequence is established. [ABSTRACT FROM AUTHOR] - Published
- 2018
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