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Matrices All of Whose Principal Submatrices of Some Order Have a Nonzero Permanent
Brualdi, Richard A.,Hwang, Suk-Egun 경북대학교 위상수학 기하학연구센터 1995 硏究論文集 Vol.5 No.-
We determine the minimum number of 1s in an (irreducible) (0, 1) matrix of order n such that for some integer k, all principal submatrices of order k, but not all of order k - 1, have a positive permanent. We also characterize the extremal matrices.
n the spectral radius of (0, 1)-matrices with 1's in prescribecl positions
Brualdi, Richard A.,Huang, Suk-Eeun 경북대학교 위상수학 기하학연구센터 1995 硏究論文集 Vol.5 No.-
Let n and d be positive integers with 1≤d≤n(n-1)/2. We investigate the maximum and minimum spectral radius of a (0,1)-matrix of order n which has 1's on and below its main diagonal and d additional 1's. If d≤4 we determine all matrices of this type which have the maximum spectral radius. For general d we prove an asymptotic results that severely limits the structure of matrices with maximum spectral radius. For d≤n, we determine the minimum spectral radius.
Vector Majorization via Hessenberg Matrices
Brualdi, Richard A.,Hwang, Suk-Eeun 경북대학교 위상수학 기하학연구센터 1995 硏究論文集 Vol.5 No.-
In this paper, it is proved that, for real n-vectors x and y, x is majorized by y if and only if x=PHQy for some permutation matrices P, Q and for some doubly stochastic matrix H which is a direct sum of doubly stochastic Hessenberg matrices. This result reveals that any n-vector which is majorized by a vector y can be expressed as a convex combination of at most (n^2 - n + 2) / 2 permutations of y.
Vector Majorization via Positive Definite Matrices
Brualdi, Richard A.,Hwang, Suk-Geun,Pyo, Sung-Soo 경북대학교 위상수학 기하학연구센터 1999 硏究論文集 Vol.8 No.-
It is well known that for real n-vectors y and x, y majorizes x if and only if Ay=x for some doubly stochastic matrix A of order n. If the components of each of y and x are in nonincreasing order, then it is known that the matrix A can be chosen to be positive semidefinite symmetric. We characterize when there is a positive definite doubly stochastic matrix A such that Ay=x. ⓒ Elsevier Science Inc., 1997
GENERALIZED ALTERNATING SIGN MATRICES AND SIGNED PERMUTATION MATRICES
Brualdi, Richard A.,Kim, Hwa Kyung Korean Mathematical Society 2021 대한수학회지 Vol.58 No.4
We continue the investigations in [6] extending the Bruhat order on n × n alternating sign matrices to our more general setting. We show that the resulting partially ordered set is a graded lattice with a well-define rank function. Many illustrative examples are given.