The signed enhanced principal rank characteristic sequence
Abstract
The signed enhanced principal rank characteristic sequence (sepr-sequence) of an n × n Hermitian matrix is the sequence t1t2 ·s tn, where tk is either A*, A+, A-, N, S*, S+, or S- based on the following criteria: tk = A* if B has both a positive and a negative order-k principal minor, and each order-k principal minor is nonzero. tk = A+ (respectively, tk = A-) if each order-k principal minor is positive (respectively, negative). tk = N if each order-k principal minor is zero. tk = S* if B has each a positive, a negative, and a zero order-k principal minor. tk = S+ (respectively, tk = S-) 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 the ( A,N,S) epr-sequence in the literature, where the kth term is either A, N, or S 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 as A*N and NA* are prohibited in the sepr-sequence of a Hermitian matrix. For Hermitian matrices of orders n=1,2,3, all attainable sepr-sequences are classified. For real symmetric matrices, a complete characterization of the attainable sepr-sequences whose underlying epr-sequence contains ANA as a non-terminal subsequence is established.
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