R. A. BRUALDI and H. J. RYSER. Combinatorial Matrix Theory. Cambridge University Press, New York, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of some special circulant matrices (see, e.g. Mi85] Mi87] MSS69] KP69] Note that the computation of the permanent of certain Toeplitz and in particular circulant matrices has applications to a number of combinatorial enumeration problems, e.g. the famous Rencontres and M enage problems [Mi78,BR91]. This paper contains an analysis of the permanents of the most sparse Toeplitz matrices for which the problem is already non trivial. These matrices include some examples of circulants for which none of the previous approaches could be successfully employed. For instance, we find efficient ....

R.A. Brualdi, and H.J. Ryser. Combinatorial matrix theory. Cambridge University Press (1991). 31

....of classifying all such primes. 2.1 Definitions Positive matrices may be regarded either as matrices, as elements of a semi ring, or as convex polyhedral cones. The relationships between the matrix, the algebraic, and the geometric approach is extremely useful. Sources on positive matrices are [2, 3, 5, 17]. In this paper the set R = 0; 1) is called the set of positive real numbers and (0; 1) the set of strictly positive real numbers. This terminology is used in [9, 2.2] Let Z = f1; 2; g denote the set of the positive integers and N = f0; 1; g the set of the natural numbers. For n 2 ....

R.A. Brualdi and H.J. Ryser. Combinatorial matrix theory. Cambridge University Press, Cambridge, 1991.

....polynomial of a matrix, and the product of a sequence of matrices [20] Many graph theory problems are also reduced to matrix multiplication. Examples are finding the transitive closure, all pairs shortest paths, the minimum weight spanning tree, topological sort, and critical paths of a graph [6], 12] Therefore, fast and processor efficient parallel algorithms for matrix multiplication is definitely of fundamental importance. The standard matrix multiplication algorithm takes O(N 3 ) operations. Most existing parallel algorithms are parallelizations of the standard method. For ....

# R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory. New York: Cambridge Univ. Press, 1991.

....Y 2 Phi(X : K) if and only if the (i; j) entry of Y is zero whenever the (i; j) entry of X is zero. 2 Permutation and doubly stochastic matrices In this section, we characterize those linear maps T on span(S) satisfying T (S) S for S = P(n) or DS(n) By the Birkhoff Theorem (e.g. see [3]) we see that E(DS(n) P(n) One easily checks that span(DS(n) span(P(n) is the subspace of n Theta n real matrices with equal row sums and column sums, and has dimension (n Gamma 1) 2 1. By Proposition 1.2, we have the following. Lemma 2.1 Let A = a ij ) 2 DS(n) Then B = b ij ) 2 ....

R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, Cambridge, 1991. 16

....i #= j and (i, j) # E, 0 otherwise. Theorem 4.2 (Ingleton and Pi#) Let G be a digraph, and let # F be its free adjacency matrix. Then for any k vertex set X and any k vertex set Y , p(X, Y ) k i# det # F (Y , X) is not identically zero. We also need a matrix identity of Jacobi (see [BR 91, Lemma 9.2.10] Fact (Jacobi) If a matrix F is nonsingular, then a square submatrix F (Y , X) is nonsingular i# the complementary submatrix F 1 (X, Y ) is nonsingular. More precisely, det(F 1 (X, Y ) det(F (Y , X) det(F ) To apply the above theorem to the algorithm, the variables ....

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, New York, 1991.

....then A contains at least one full 2 Theta 2 submatrix with no entry on the main diagonal. Proof. If A does not have total support, then there exist two permutation matrices P and Q such that B = PAQ is a block triangular matrix, whose diagonal blocks B 1 ; B 2 ; B t have total support [3]. The matrices B i are uniquely determined within arbitrary permutation of their rows, but their ordering in B is not necessarily unique. As a consequence we cannot guarantee that the multiplication by P and Q maps all the diagonal elements of A inside the union of the blocks B i . However, we can ....

R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, 1991, Cambridge University Press, Cambridge.

....Let X 1 ; X p denote p non commuting indeterminates and define S p = M oe even X oe(1) X oe(p) S Gamma p = M oe odd X oe(1) X oe(p) where the sums are taken over the even and odd permutations of f1; pg. The well known AmitsurLevitski theorem (see e.g. [6]) states that the polynomial identity S 2n = S Gamma 2n holds in the matrix ring R n Thetan (where R denotes a commutative ring) Such combinatorial identities extend to semirings: 4.3.1 Lemma Let A be a commutative semiring. Then, the identity S 2n = S Gamma 2n holds in A ....

R. R. Brualdi and H. J. Ryser. Combinatorial Matrix Theory. Cambridge University Press, 1991.

....matrix, and P r and P c are permutation matrices. In the CCF, each diagonal block is a full rank square matrix. For a singular or rectangular matrix the CCF includes also full rank horizontal and or vertical tails. Note that the CCF reduces to the well known Dulmage Mendelsohn decomposition [BR91, DER86, DM59, DR78, EGLPS87, Gu76, Ho76, PF90] if the Q part is empty. The example below illustrates an LM matrix and its CCF. Example 1. If t i (i = 1; 2; 3; 4) denote independent parameters, A = Q T = 0 B B B 1 1 1 1 0 0 2 1 1 0 t 1 0 0 0 t 2 0 t 3 0 0 t 4 1 C C C A is a 4 2 5 LM matrix. The CCF of A is given by A = 0 ....

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, London, 1991.

....digraph are also provided. Key words. totally positive matrix, graph, digraph, nonintersecting paths AMS subject classifications. 15A48, 05C20, 05C50 PII. S089547989630396X 1. Introduction. The relations between graph theory and matrix theory constitute a well established area of research (see [7]) This paper explores some connections between theoretic properties of totally positive matrices and graph theoretic properties of certain graphs naturally associated with the matrices. Section 2 deals with undirected graphs. Given a symmetric matrix A = a ij ) 1#i,j#n , the undirected graph ....

R. A. BRUALDI AND H. J. RYSER, Combinatorial Matrix Theory, Encyclopedia Math. Appl. 39, Cambridge University Press, Cambridge, 1991.

.... a and b such that akAk jAj bkAk for all A in a matrix space equipped with two norms k Delta k and j Delta j, 47,79] applying the theory of matrix inequalities to perturbation theory, and other applied subjects such as crystallography and game theory, 70,97,110,b,d] One may see [Bh] [BrR], HLP] HoJ, Chapter 3] SS] and the website Matrix inequalities in Science and Engineering (http: www.wm.edu CAS MINEQ matrix.html) for more specific questions, topics, research groups, and other links. We are most interested in developing and applying the theory of matrix inequalities to ....

R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, New York, 1991.

....combination of A and I . And moreover, we don t need regularity if the Laplacian matrix is used. If D is the diagonal matrix containing the vertex degrees of #, then F = D A is the Laplacian matrix. It easily follows that F is positive semi definite with row sum # 1 (F ) 0; see for example [1]. Corollary 4.5 . Suppose F is the Laplacian matrix of # (# non empty) Then # # (#) # v 2 1 # min # (F ) # max # (F ) Proof. Define M # = 2 # 2 #v F I.ThenM # #M # # and # abs # (M # ) # v # 2 ) # v # 2 ) and the result follows. 2 Suppose # is the complete ....

R.A. Brualdi and H.J. Ryser, Combinatorial matrix theory, Cambridge University Press, Cambridge, 1991.

....the strong Hall property implies the Hall property. If the Hall property is a linear independence condition, the strong Hall property is an irreducibility condition: any matrix that is not strong Hall can be permuted to a block upper triangular form called the Dulmage Mendelsohn decomposition [3, 24, 29], in which each diagonal block is strong Hall. 3 Linear equation systems and leastsquares problems whose matrices are not strong Hall can be solved by performing first a Dulmage Mendelsohn decomposition, and then a block backsubstitution that solves a system with each strong Hall diagonal block. ....

....if and only if it has no independent set of at least m vertices that includes at least one vertex from each part. A square strong Hall matrix is often called fully indecomposable, meaning that there is no way to permute its rows and columns into a block triangular form with more than one block [3]. This gives the following (standard) result. Theorem 2.6. Let H = H(A) be a square strong Hall graph. Then for all row and column permutations P r and P c , the directed graph G(P r AP c ) is strongly connected. We conclude this subsection by proving a theorem (Theorem 2.9) about strong ....

Richard A. Brualdi and Herbert J. Ryser. Combinatorial Matrix Theory. Cambridge University Press, 1991.

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R. A. BRUALDI and H. J. RYSER. Combinatorial Matrix Theory. Cambridge University Press, New York, 1991.

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R. Brualdi and H. Ryser. Combinatorial Matrix Theory. Cambridge University Press, 1991.

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R. A. Brualdi and H. J. Ryser. Combinatorial Matrix Theory. Cambridge University Press, Cambridge, 1991.

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R. Brualdi and H. Ryser. Combinatorial Matrix Theory. Cambridge University Press, 1991.

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R. A. Brualdi and H. Ryser, Combinatorial Matrix Theory. Cambridge University Press, Cambridge, 1991.

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R.A. Brualdi, and H.J. Ryser. Combinatorial matrix theory. Cambridge University Press (1991).

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R. A. BRUALDI and H. J. RYSER. Combinatorial Matrix Theory. Cambridge University Press, New York, 1991.

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R.A. Brualdi, and H.J. Ryser. Combinatorial matrix theory. Cambridge University Press (1991).

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R. A. Brualdi, H. Ryser, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1991.

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R. A. BRUALDI and H. J. RYSER. Combinatorial Matrix Theory. Cambridge University Press, New York, 1991.

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R. Brualdi and H. Ryser. Combinatorial Matrix Theory. Cambridge University Press, Cambridge, 1991.

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R. A. Brualdi and H. J. Ryser. Combinatorial matrix theory. Cambridge University Press, Cambridge, 1991.

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R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, Cambridge University Press (1991).

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