Peter M. Gibson

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

Abstract The permanent function is used to determine geometrical properties of the set Ω n of all n × n nonnegative doubly stochastic matrices. If F is a face of Ω n , then F corresponds to an n × n (0, 1)-matrix A, where the permanent of A is the number of vertices of F . If A is fully indecomposable, then the dimension of F equals σ(A) − 2n + 1, where σ(A) is the number of 1s in A. The only two-dimensional faces of Ω n are triangles and rectangles. For n ⩾ 6, Ω n has four types of three-dimensional faces. The facets of the faces of Ω n are characterized. Faces of Ω n which are simplices are determined. If F is a face of Ω n which is two-neighborly but not a simplex, then F has dimension 4 and six vertices. All k-dimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2k. All k-dimensional faces with at least 2k−1 + 1 vertices are determined.

Convex polyhedra of doubly stochastic matrices—IV

Abstract Basic geometrical properties of general convex polyhedra of doubly stochastic matrices are investigated. The faces of such polyhedra are characterized, and their dimensions and facets are determined. A connection between bounded faces of doubly stochastic polyhedra and faces of transportation polytopes is established, and it is shown that there exists an absolute bound for the number of extreme points of d -dimensional bounded faces of these polyhedra.

Convex polyhedra of doubly stochastic matrices: II. Graph of Ωn

Abstract Properties of the graph G(Ω n ) of the polytope Ω n of all n × n nonnegative doubly stochastic matrices are studied. If F is a face of Ω n which is not a k -dimensional rectangular parallelotope for k ≥ 2, then G ( F ) is Hamilton connected. Prime factor decompositions of the graphs of faces of Ω n relative to Cartesian product are investigated. In particular, if F is a face of Ω n , then the number of prime graphs in any prime factor decomposition of G ( F ) equals the number of connected components of the neighborhood of any vertex of G ( F ). Distance properties of the graphs of faces of Ω n are obtained. Faces F of Ω n for which G ( F ) is a clique of G(Ω n ) are investigated.

Convex polyhedra of doubly stochastic matrices III. Affine and combinatorial properties of Ωn

Abstract Affine and combinatorial properties of the polytope Ω n of all n × n nonnegative doubly stochastic matrices are investigated. One consequence of this investigation is that if F is a face of Ω n of dimension d > 2, then F has at most 3( d −1) facets. The special faces of Ω n which were characterized in Part I of our study of Ω n in terms of the corresponding (0, 1)- matrices are classified with respect to affine equivalence.

Simultaneous diagonalization of rectangular complex matrices

Abstract A number of necessary and sufficient conditions are given for the existence of unitary matrices U and V , such that UAV is a diagonal matrix for every matrix A in some set Γ of rectangular complex matrices. Two related questions are then considered. A necessary and sufficient condition for the existence of unitary matrices U and V such that UAV is a real diagonal matrix for every A in Γ is obtained, and an improvement on a necessary and sufficient condition discovered by R.C. Thompson for the existence of real orthogonal matrices P and Q such that PAQ is a diagonal matrix for every A in Γ is given.

Combinatorially orthogonal matrices and related graphs

Abstract Let G be a graph and let c ( x , y ) denote the number of vertices in G adjacent to both of the vertices x and y . We call G quadrangular if c ( x , y ) ≠ 1 whenever x and y are distinct vertices in G . Reid and Thomassen proved that | E ( G )| ⩾ 2| V ( G )| −4 for each connected quadrangular graph G , and characterized those graphs for which the lower bound is attained. Their result implies lower bounds on the number of 1s in m × n combinatorially orthogonal (0,1)-matrices, where a (0,1)-matrix A is said to be combinatorially orthogonal if the inner product of each pair of rows and each pair of columns of A is never one. Thus the result of Reid and Thomassen is related to a paper of Beasley, Brualdi and Shader in which they show that a fully indecomposable, combinatorially orthogonal (0,1)-matrix of order n ⩾ 2 has at least 4 n − 4 ls and characterize those matrices for which equality holds. One of the results obtained here is equivalent to the result of Beasley, Brualdi and Shader. We also prove that | E ( G )| ⩾ 2| V ( G )| − 1 for each connected quadrangular nonbipartite graph G with at least 5 vertices, and characterize the graphs for which the lower bound is attained. In addition, we obtain optimal lower bounds on the number of 1s in m × n combinatorially row-orthogonal (0,1)-matrices.

Permanental Polytopes of Doubly Stochastic Matrices

Abstract A subpolytope Γ of the polytope Ωn of all n×n nonnegative doubly stochastic matrices is said to be a permanental polytope if the permanent function is constant on Γ. Geometrical properties of permanental polytopes are investigated. No matrix of the form 1⊕A where A is in Ω2 is contained in any permanental polytope of Ω3 with positive dimension. There is no 3-dimensional permanental polytope of Ω3, and there is essentially a unique maximal 2-dimensional permanental polytope of Ω3 (a square of side 1 3 ). Permanental polytopes of dimension (n 2 −3n+4) 2 are exhibited for each n⩾4.

The assignment polytope

The a s s i g n m e n t p o l y t o p e g~, cons i s t s of all n x n n o n n e g a t i v e d o u b l y s tochas t i c ma t r i c e s , tha t is, n x n rea l ma t r i ce s X = [x~j] such tha t x,j >t 0 and Z~=~ x~k = 1 = Z~=l x~ fo r all i, j = 1 . . . . . n. I t is wel l k n o w n tha t the d i m e n s i o n of On r e g a r d e d as a subse t of rea l n2-space equa ls ( n l ) 2 and tha t the n x n p e r m u t a t i o n m a t r i c e s a re the ve r t i c e s of g],. Ba l in sk i and R u s s a k o f f [1, 2] have r e c e n t l y d e t e r m i n e d a n u m b e r of p r o p e r t i e s of O,. In this r e s e a r c h a n n o u n c e m e n t we s ta te some o the r resu l t s fo r O,. P r o o f s of t he se and r e l a t ed resu l t s wil l a p p e a r in [3, 4, 5]. L e t A = [a~j] be an n x n (0, 1)-matr ix. I f n > 1, t hen A is fully indecomposable p r o v i d e d the re does no t ex i s t an in teger r wi th 1 ~< r ~< n 1 such tha t A con ta ins an r x (n r ) ze ro subma t r ix . I f n = 1, then A is fu l ly i n d e c o m p o s a b l e if and on ly if A = [1]. The ma t r i x A is said to have total support if t he re ex i s t p e r m u t a t i o n ma t r i ce s P and Q such tha t P A Q is a d i r ec t sum A ~ @ . . . @ A t w h e r e A~ . . . . . A, a re fu l ly i n d e c o m p o s a b l e . The m a t r i c e s A1 . . . . . A, w h i c h have o r d e r at l eas t 2 a re ca l led non-trivial fully indecomposable components of A. I t is wel l k n o w n tha t the ma t r i x A has to ta l s u p p o r t if and on ly if A ¢ 0 and ar~ = 1 impl ies tha t the re ex is t s a p e r m u t a t i o n m a t r i x P = [p,j] w i th p~ = 1 and P ~ A. L e t ~ be a n o n e m p t y f a c e of S2,. T h e n the re ex i s t s a un ique n x n (0, 1)-matr ix A wi th to ta l s u p p o r t such tha t cons i s t s of all m a t r i c e s X in g2, for w h i c h X ~< A. W e wr i t e ~ = o%(A).

Generalized doubly stochastic and permutation matrices over a ring

Abstract Let R be a ring with unity. A combinatorial argument is used to show that the R -module Δ n ( R ) of all n × n matrices over R with constant row and column sums has a basis consisting of permutation matrices. This is used to characterize orthogonal matrices which are linear combinations of permutation matrices. It is shown that all bases of Δ n ( R ) consisting of permutation matrices have the same cardinality, and other properties of bases of Δ n ( R ) are investigated.

Permanents of d-dimensional matrices

Let A = (Ai1i2…id) be an n1 × n2 × · × nd matrix over a commutative ring. The permanent of A is defined by per (A) = ∑πn1i = 1Aiσ2(i)σ3(i)…σd(i), where the summation ranges over all one-to-one functions σk from {1,2,…, n1} to {1,2,…, nk}, k = 2,3,…, d. In this paper it is shown that a number of properties of permanents of 2-dimensional matrices extend to higher-dimensional matrices. In particular, permanents of nonnegative d-dimensional matrices with constant hyperplane sums are investigated. The paper concludes by introducing s-permanents, which generalize the definition above that the permanent becomes the 1-permanent, and showing that an s-permanent can always be converted into a 1-permanent.