Showing papers by "Richard A. Brualdi published in 1994"
TL;DR: The Gersgorin circle theorem as discussed by the authors is a simple analytic theorem involving elementary inequalities derived from the basic algebraic eigenvalue/eigenvector equation, which is one of the rare instances of a theorem which is elegant and useful and which has a short, elementary proof.
Abstract: The Gersgorin circle theorem gives a region in the complex plane which contains all the eigenvalues of a square complex matrix. It is one of those rare instances of a theorem which is elegant and useful and which has a short, elementary proof. It is surprising that it hasn't made its way into many introductory texts on linear algebra. One reason for this neglect may be the fact that many (even most) mathematicians still regard linear algebra as being only about algebra. But modern linear algebra is more than algebra. It's linear systems, matrix theory and analysis, geometry, applications, numerics and, of course, algebra. The first course in linear algebra at most colleges and universities is to a great extent a service course for future scientists. Gersgorin's theorem is not an algebraic theorem. It is a simple analytic theorem involving elementary inequalities derived from the basic algebraic eigenvalue/ eigenvector equation. If A = [aij] is a complex matrix of order n and A is an eigenvalue of A, then there exists a nonzero vector x = (x1, x2,. . ., xn)T in Cn such that
78 citations
TL;DR: Sign-Central matrices were introduced and given a combinatorial characterization by Davydov and Davydova as mentioned in this paper, who gave an alternative proof of their characterization and showed that under a minimality assumption a sign-central matrix with m nonzero rows has at least m + 1 columns, and that equality holds if and only if the matrix is an S-matrix (as defined in the theory of sign solvability).
21 citations
TL;DR: This paper introduces two new classes of L -matrices, which for square matrices reduce to sign-nonsingular matrices, and the maximum number of columns for matrices in each of these classes is obtained, and those matrices attaining the maximum are characterized.
17 citations
TL;DR: This work considers the poset of all posets on n elements where the partial order is that of inclusion of comparabilities and discusses some properties of this poset concerning its height, width, jump number and dimension.
11 citations
TL;DR: This characterization is used to determine the sign patterns of the inverses of fully indecomposable, strong sign-nonsingular matrices and to develop a recognition algorithm for such sign patterns.
11 citations
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TL;DR: In this article, the authors developed an inductive structure for nonsquare strong Hall matrices that is quite analogous to the well-known induction structure of square strong Hall (i.e., fully indecomposable) matrices.
Abstract: The authors develop an inductive structure for nonsquare strong Hall matrices that is quite analogous to the well-known inductive structure of square strong Hall (i.e., fully indecomposable) matrices. Other properties of strong Hall matrices are also discussed.
8 citations
TL;DR: In this article, the minimum permanent on generalized Hessenberg faces of the polytope of doubly stochastic matrices was determined and the sub-problems of matrices achieving the minimum were investigated.
7 citations
TL;DR: The maximum number of edges of a graph of order $n$ that does not contain neither $K_t$ nor $K_{t,t}$ as a subgraph is determined.
Abstract: We consider the following two problems. (1) Let $t$ and $n$ be positive integers with $n\geq t\geq 2$. Determine the maximum number of edges of a graph of order $n$ that contains neither $K_t$ nor $K_{t,t}$ as a subgraph. (2) Let $r$, $t$ and $n$ be positive integers with $n\geq rt$ and $t\geq 2$. Determine the maximum number of edges of a graph of order $n$ that does not contain $r$ disjoint copies of $K_t$. Problem 1 for $n < 2t$ is solved by Turan's theorem and we solve it for $n=2t$. We also solve Problem 2 for $n=rt$.
2 citations
TL;DR: In this paper, the authors introduced a class of n×n matrices of 0's and 1's which can be regarded as generalizations of the classical Ferrers boards of rook theory.
Abstract: We introduce a class ofn×n matrices of 0’s and 1’s which can be regarded as generalizations of the classical Ferrers boards of rook theory. Assuming the matrices are fully indecomposable, we determine the minimum permanent and the minimum number of 1’s as a function ofn. We also characterize these matrices in terms of weighted, top-rooted trees.
1 citations