A fiedler-type theorem for the determinant of J-positive matrices

https://thekeep.eiu.edu/commencement_spring2014/1030/thumbnail.jp

Reciprocal matrices: properties and approximation by a transitive matrix

N. Bebiano: partially supported by project UID/MAT/00324/2019. R. Fernandes: partially supported by project UID/MAT/00297/2019. S. Furtado: partially supported by project UID/MAT/04721/2019.Reciprocal matrices and, in particular, transitive matrices, appear in several applied areas. Among other applications, they have an important role in decision theory in the context of the analytical hierarchical process, introduced by Saaty. In this paper, we study the possible ranks of a reciprocal matrix and give a procedure to construct a reciprocal matrix with the rank and the off-diagonal entries of an arbitrary row (column) prescribed. We apply some techniques from graph theory to the study of transitive matrices, namely to determine the maximum number of equal entries, and distinct from ± 1 , in a transitive matrix. We then focus on the n-by-n reciprocal matrix, denoted by C(n, x), with all entries above the main diagonal equal to x> 0. We show that there is a Toeplitz transitive matrix and a transitive matrix preserving the maximum possible number of entries of C(n, x), whose distances to C(n, x), measured in the Frobenius norm, are smaller than the one of the transitive matrix proposed by Saaty, constructed from the right Perron eigenvector of C(n, x). We illustrate our results with some numerical examples.authorsversionpublishe

On the corners of certain determinantal ranges

Let A be a complex n×n matrix and let SO(n) be the group of real orthogonal matrices of determinant one. Define [Delta](A)={det(AoQ):Q[set membership, variant]SO(n)}, where o denotes the Hadamard product of matrices. For a permutation [sigma] on {1,...,n}, define It is shown that if the equation z[sigma]=det(AoQ) has in SO(n) only the obvious solutions (Q=([epsilon]i[delta][sigma]i,j), [epsilon]i=±1 such that [epsilon]1...[epsilon]n=sgn[sigma]), then the local shape of [Delta](A) in a vicinity of z[sigma] resembles a truncated cone whose opening angle equals , where [sigma]1, [sigma]2 differ from [sigma] by transpositions. This lends further credibility to the well known de Oliveira Marcus Conjecture (OMC) concerning the determinant of the sum of normal n×n matrices. We deduce the mentioned fact from a general result concerning multivariate power series and also use some elementary algebraic topology.http://www.sciencedirect.com/science/article/B6V0R-4NJG44V-3/1/29cc71d6352bcfea422c3dc7beebcbc

Convexity of the Krein space tracial numerical range and Morse theory

In this paper we present a Krein space convexity theorem on the tracial-numerical range of a matrix. This theorem is the analogue of Westwick's theorem. The proof is an application of Morse theory

"McDonaldizar" a Matemática da geração Nintendo?

Na sociedade global, em particular na sociedade portuguesa, o preconceito contra a matemática assume contornos merecedores de reflexão. Ciência morta, onde tudo está feito, ciência elitista ou imperialista, com exagerado peso selectivo nos curricula e no ingresso no ensino superior, são alguns epítetos que exornam a sua imagem. Existe uma subavaliação da matemática por parte da opinião pública. Cabe aos matemáticos a promoção de uma nova relação da sociedade com a sua ciência. A matemática tem lugar central nos currículos escolares em todo o mundo e, em todo o mundo, milhares de matemáticos ocupam-se da resolução de problemas extremamente variados. Milhares de matemáticos e de pedagogos inventam técnicas inovadoras e novas modalidades de ensino, tentando fomentar o sucesso numa disciplina com inegável peso no acesso à maioria dos cursos superiores