Suk-Geun Hwang

The inverse eigenvalue problem for symmetric doubly stochastic matrices

Abstract For a positive integer n and for a real number s, let Γns denote the set of all n×n real matrices whose rows and columns have sum s. In this note, by an explicit constructive method, we prove the following. (i) Given any real n-tuple Λ=(λ1,λ2,…,λn)T, there exists a symmetric matrix in Γnλ1 whose spectrum is Λ. (ii) For a real n-tuple Λ=(1,λ2,…,λn)T with 1⩾λ2⩾⋯⩾λn, if 1 n + λ 2 n(n−1) + λ 3 n(n−2) +⋯+ λ n 2·1 ⩾0, then there exists a symmetric doubly stochastic matrix whose spectrum is Λ. xa0The second assertion enables us to show that for any λ2,…,λn∈[−1/(n−1),1], there is a symmetric doubly stochastic matrix whose spectrum is (1,λ2,…,λn)T and also that any number β∈(−1,1] is an eigenvalue of a symmetric positive doubly stochastic matrix of any order.

Matrices determined by a linear recurrence relation among entries

Abstract A matrix A =[ a ij ] is called a 7-matrix if its entries satisfy the recurrence relation αa i −1, j −1 + βa i −1, j = a ij where α , β are fixed numbers. A 7-matrix is completely determined by its first row and first column. In this paper we determine the structure of 7-matrices and investigate the sequences represented by columns of infinite 7-matrices.

Matrix majorization via vector majorization

For two real m×n matrices X and Y, Y is said to majorize X if SY=X for some doubly stochastic matrix S of order m. In this note, we prove that Y majorizes X if and only if Yv majorizes Xv for every real n-vector v, under the assumption that [X,e][Y,e]+ is nonnegative, where e and [Y,e]+ denote the m-vector of ones and the Moore–Penrose generalized inverse of [Y,e], respectively.

Permanents of doubly stochastic trees

Abstract For a tree T of order n , let Ω(T)={X∈Ω n ∣X⩽A(T)+I n } , where Ω n denotes the set of all doubly stochastic matrices of order n and A ( T ) denotes the adjacency matrix of T , and let μ ( T ) denote the minimum permanent of matrices in Ω(T) . Let P n denote the path of length n −1 and K 1, n −1 the complete bipartite graph on 1+( n −1) vertices. In this paper, it is shown that P n and K 1, n −1 are the only trees with minimal and maximal μ -values respectively among all trees of order n .

An upper bound for the permanent of a nonnegative matrix

Abstract Let A be a fully indecomposable, nonnegative matrix of order n with row sums r l , r n , and let s i equal the smallest positive element in row i of A . We prove the permanental inequality per (A)⩽ ∏ i=1 n s i + ∏ i=1 n (r i −s i ) and characterize the case of equality. In 1984 Donald, Elwin, Hager, and Salamon gave a graph-theoretic proof of the special case in which A is a nonnegative integer matrix.

Maximum permanents on certain classes of nonnegative matrices

Abstract Let UR(α, β) denote the class of all square matrices with each entry equal to one of the nonnegative numbers α and β and with row sum vector R. We prove that the maximum value of the permanent of a matrix in the convex hull of UR(α, β) is achieved at a matrix in UR(α, β). One consequence of this result is an extension to matrices with entries between 0 and 1 of the Minc-Bregman upper bound for permanents of (0,1) matrices in terms of the integral row sums. We obtain other upper bounds for permanents of certain nonnegative matrices and give a nonprobabilistic proof of an inequality of Chang for the permanents of certain doubly stochastic matrices.

Doubly stochastic matrices whose powers eventually stop

Abstract In this note we characterize doubly stochastic matrices A whose powers A,A 2 ,A 3 ,… eventually stop, i.e., A p =A p+1 =⋯ for some positive integer p. The characterization enables us to determine the set of all such matrices.

MINIMIZING THE PERMANENT OVER SOME FACES OF THE POLYTOPE OF DOUBLY STOCHASTIC MATRICES

Abstract We determine the minimum permanents and minimizing matrices over certain faces of the polytope of doubly stochastic matrices.

A Real Proof of the Principal Axis Theorem

Summary In this capsule we give an elementary proof of the principal axis theorem within the real field, i.e., without using complex numbers.

Majorization via generalized Hessenberg matrices

Abstract In this paper we give a method of constructing generalized Hessenberg matrices from those of smaller orders, with a simple combinatorial justification. Making use of this construction, we also prove that, for real n -vectors x and y whose components are arranged in nonincreasing order, x is majorized by y if and only if there exists a doubly stochastic matrix A of order n with x =A y whose support is permutation similar to a direct sum of generalized Hessenberg matrices.