The workshop brought together mathematicians with a common interest in nonnegativity as it arises... more The workshop brought together mathematicians with a common interest in nonnegativity as it arises in linear algebra, operator theory and max algebra. The goals included making progress on important open problems, identifying new directions and challenges, developing global themes that tie notions of nonnegativity together, as well as exchanging and comparing ideas in the study of nonnegativity of linear and non-linear maps.
In this paper it is shown that an eventually nonnegative matrix A whose index of zero is less tha... more In this paper it is shown that an eventually nonnegative matrix A whose index of zero is less than or equal to one, exhibits many of the same combinatorial properties as a nonnegative matrix. In particular, there is a positive integer g such that Ag is nonnegative, A and Ag have the same irreducible classes, and the transitive closure of the reduced graph of A is the same as the transitive closure of the reduced graph of Ag. In this instance, many of the combinatorial properties of nonnegative matrices carry over to this subclass of the eventually nonnegative matrices. AMS subject classifications. 15A18, 15A48, 15A21
The workshop Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns, he... more The workshop Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns, held at the American Institute of Mathematics Research Conference Center on Oct. 23-27, 2006, focused on three problems: • Determination of the minimum rank, or equivalently maximum multiplicity of an eigenvalue, of real symmetric matrices described by a graph. • The 2n-conjecture for spectrally arbitrary sign patterns. • The energy of graphs.
An nxn matrix pattern is said to be spectrally arbitrary over a field F provided for every monic ... more An nxn matrix pattern is said to be spectrally arbitrary over a field F provided for every monic polynomial p(t) of degree n, with coefficients from F, there exists a matrix with entries from F, in the given pattern, that has characteristic polynomial p(t). Let E subset of F subset of K be an extension of fields. It is natural to ask whether a pattern that is spectrally arbitrary over F must also be spectrally arbitrary over E or K. In this article it is shown that if F is dense in K and K is a complete metric space, then any spectrally arbitrary or relaxed spectrally arbitrary pattern over F is relaxed spectrally arbitrary over K. It is also established that if E is an algebraically closed subfield of a field F, then any spectrally arbitrary pattern over F is spectrally arbitrary over E. The 2n Conjecture and the Superpattern Conjecture are explored over fields other than the real numbers. In particular, examples are provided to show that the Superpattern Conjecture is false over the field with 3 elements.
An n by n zero–nonzero pattern is a matrix with entries ∈{*, 0} where * denotes a nonzero real nu... more An n by n zero–nonzero pattern is a matrix with entries ∈{*, 0} where * denotes a nonzero real number. If allows all possible inertias, then is inertially arbitrary. It is shown that there exists a reducible n by n inertially arbitrary zero–nonzero pattern with 2n−1 nonzero entries for each n ≥ 6; and that for n = mt with t ≥ 6 and m ≥ 1, there
SIAM Journal on Matrix Analysis and Applications, 2004
... Minimal spectrally arbitrary sign patterns T. Britz JJ McDonald DD Olesky P. van den Driessch... more ... Minimal spectrally arbitrary sign patterns T. Britz JJ McDonald DD Olesky P. van den Driessche ... Page 2. Page 3. MINIMAL SPECTRALLY ARBITRARY SIGN PATTERNS∗ T. BRITZ†, JJ MCDONALD‡, DD OLESKY§, AND P. VAN DEN DRIESSCHE† Abstract. ...
The workshop brought together mathematicians with a common interest in nonnegativity as it arises... more The workshop brought together mathematicians with a common interest in nonnegativity as it arises in linear algebra, operator theory and max algebra. The goals included making progress on important open problems, identifying new directions and challenges, developing global themes that tie notions of nonnegativity together, as well as exchanging and comparing ideas in the study of nonnegativity of linear and non-linear maps.
In this paper it is shown that an eventually nonnegative matrix A whose index of zero is less tha... more In this paper it is shown that an eventually nonnegative matrix A whose index of zero is less than or equal to one, exhibits many of the same combinatorial properties as a nonnegative matrix. In particular, there is a positive integer g such that Ag is nonnegative, A and Ag have the same irreducible classes, and the transitive closure of the reduced graph of A is the same as the transitive closure of the reduced graph of Ag. In this instance, many of the combinatorial properties of nonnegative matrices carry over to this subclass of the eventually nonnegative matrices. AMS subject classifications. 15A18, 15A48, 15A21
The workshop Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns, he... more The workshop Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns, held at the American Institute of Mathematics Research Conference Center on Oct. 23-27, 2006, focused on three problems: • Determination of the minimum rank, or equivalently maximum multiplicity of an eigenvalue, of real symmetric matrices described by a graph. • The 2n-conjecture for spectrally arbitrary sign patterns. • The energy of graphs.
An nxn matrix pattern is said to be spectrally arbitrary over a field F provided for every monic ... more An nxn matrix pattern is said to be spectrally arbitrary over a field F provided for every monic polynomial p(t) of degree n, with coefficients from F, there exists a matrix with entries from F, in the given pattern, that has characteristic polynomial p(t). Let E subset of F subset of K be an extension of fields. It is natural to ask whether a pattern that is spectrally arbitrary over F must also be spectrally arbitrary over E or K. In this article it is shown that if F is dense in K and K is a complete metric space, then any spectrally arbitrary or relaxed spectrally arbitrary pattern over F is relaxed spectrally arbitrary over K. It is also established that if E is an algebraically closed subfield of a field F, then any spectrally arbitrary pattern over F is spectrally arbitrary over E. The 2n Conjecture and the Superpattern Conjecture are explored over fields other than the real numbers. In particular, examples are provided to show that the Superpattern Conjecture is false over the field with 3 elements.
An n by n zero–nonzero pattern is a matrix with entries ∈{*, 0} where * denotes a nonzero real nu... more An n by n zero–nonzero pattern is a matrix with entries ∈{*, 0} where * denotes a nonzero real number. If allows all possible inertias, then is inertially arbitrary. It is shown that there exists a reducible n by n inertially arbitrary zero–nonzero pattern with 2n−1 nonzero entries for each n ≥ 6; and that for n = mt with t ≥ 6 and m ≥ 1, there
SIAM Journal on Matrix Analysis and Applications, 2004
... Minimal spectrally arbitrary sign patterns T. Britz JJ McDonald DD Olesky P. van den Driessch... more ... Minimal spectrally arbitrary sign patterns T. Britz JJ McDonald DD Olesky P. van den Driessche ... Page 2. Page 3. MINIMAL SPECTRALLY ARBITRARY SIGN PATTERNS∗ T. BRITZ†, JJ MCDONALD‡, DD OLESKY§, AND P. VAN DEN DRIESSCHE† Abstract. ...
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Papers by Judith McDonald