410 research outputs found
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Generalized Finite Algorithms for Constructing Hermitian Matrices with Prescribed Diagonal and Spectrum
In this paper, we present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of (N-1) or fewer plane rotations, where N is the dimension of the matrix. Both the Bendel-Mickey and the Chan-Li algorithms are special cases of the proposed procedures. Using the fact that a positive semidefinite matrix can always be factored as \mtx{X^\adj X}, we also provide more efficient versions of the algorithms that can directly construct factors with specified singular values and column norms. We conclude with some open problems related to the construction of Hermitian matrices with joint diagonal and spectral properties
Tridiagonal and Pentadiagonal Doubly Stochastic Matrices
We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix A is completely positive and provide examples including how one can change the initial conditions or deal with block matrices, which expands the range of matrices to which our decomposition can be applied. Our decomposition leads us to a number of related results, allowing us to prove that for tridiagonal doubly stochastic matrices, positive semidefiniteness is equivalent to complete positivity (rather than merely being implied by complete positivity). We then consider symmetric pentadiagonal matrices, proving some analogous results, and providing two different decompositions sufficient for complete positivity, again illustrated by a number of examples
Tridiagonal and Pentadiagonal Doubly Stochastic Matrices
We provide a decomposition that is sufficient in showing when a symmetric
tridiagonal matrix is completely positive and provide examples including
how one can change the initial conditions or deal with block matrices, which
expands the range of matrices to which our decomposition can be applied. Our
decomposition leads us to a number of related results, allowing us to prove
that for tridiagonal doubly stochastic matrices, positive semidefiniteness is
equivalent to complete positivity (rather than merely being implied by complete
positivity). We then consider symmetric pentadiagonal matrices, proving some
analogous results, and providing two different decompositions sufficient for
complete positivity, again illustrated by a number of examples.Comment: 15 page
Spectra, pseudospectra, and localization for random bidiagonal matrices
There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random non-hermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a "bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the non-periodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the infinite-dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of finite bidiagonal matrices, infinite bidiagonal matrices ("stochastic Toeplitz operators"), finite periodic matrices, and doubly infinite bidiagonal matrices ("stochastic Laurent operators").\ud
\ud
This is a preprint of an article published in Communications in Pure and Applied Mathematics, copyright 2000, John Wiley & Sons, Inc. This work was supported by the UK Engineering and Physical Sciences Research Council Grant GR/M1241
How long does it take to compute the eigenvalues of a random symmetric matrix?
We present the results of an empirical study of the performance of the QR
algorithm (with and without shifts) and the Toda algorithm on random symmetric
matrices. The random matrices are chosen from six ensembles, four of which lie
in the Wigner class. For all three algorithms, we observe a form of
universality for the deflation time statistics for random matrices within the
Wigner class. For these ensembles, the empirical distribution of a normalized
deflation time is found to collapse onto a curve that depends only on the
algorithm, but not on the matrix size or deflation tolerance provided the
matrix size is large enough (see Figure 4, Figure 7 and Figure 10). For the QR
algorithm with the Wilkinson shift, the observed universality is even stronger
and includes certain non-Wigner ensembles. Our experiments also provide a
quantitative statistical picture of the accelerated convergence with shifts.Comment: 20 Figures; Revision includes a treatment of the QR algorithm with
shift
A Survey of Alternating Permutations
This survey of alternating permutations and Euler numbers includes
refinements of Euler numbers, other occurrences of Euler numbers, longest
alternating subsequences, umbral enumeration of classes of alternating
permutations, and the cd-index of the symmetric group.Comment: 32 pages, 7 figure
- …