Mathematics & Statistics Faculty

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Browsing Mathematics & Statistics Faculty by Author "Butler, Steve"

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    The enhanced principal rank characteristic sequence
    (Elsevier, 2015) Butler, Steve; Catral, Minnie; Fallat, Shaun; Hall, Tracy; Hogben, Leslie; van den Driessche, Pauline; Young, michael
    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.
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    The maximum nullity of a complete edge subdivision graph is equal to it zero forcing number
    (International Linear Algebra Society, 2014-06) Barrett, Wayne; Butler, Steve; Catral, Minnie; Hall, Tracy; Fallat, Shaun; Hogben, Leslie; Young, Michael
    Barrett et al. asked in [W. Barrett et al. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530–563, 2009.], whether the maximum nullity is equal to the zero forcing number for all complete subdivision graphs. We prove that this equality holds. Furthermore, we compute the value of M(F, °G) = Z(°G) by introducing the bridge tree of a connected graph. Since this equality is valid for all fields, °G has field independent minimum rank, and we also show that °G has a universally optimal matrix.
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    The principal rank characteristic sequence over various fields
    (Elsevier, 2014) Barrett, Wayne; Butler, Steve; Catral, Minnie; Fallat, Shaun; Hall, Tracy; Hogben, Leslie; van den Driessche, Pauline; Young, Michael
    Given an n-by-n matrix, its principal rank characteristic sequence is a sequence of length n + 1 of 0s and 1s where, for k = 0; 1,..., n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such attainable sequences determined for all n over any field with characteristic 2. A complete list of attainable sequences for real symmetric matrices of order 7 is reported.