10,175 research outputs found
A regularization algorithm for matrices of bilinear and sesquilinear forms
We give an algorithm that uses only unitary transformations and for each
square complex matrix constructs a *congruent matrix that is a direct sum of a
nonsingular matrix and singular Jordan blocks.Comment: 18 page
Canonical forms for complex matrix congruence and *congruence
Canonical forms for congruence and *congruence of square complex matrices
were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353],
based on Sergeichuk's paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501],
which employed the theory of representations of quivers with involution. We use
standard methods of matrix analysis to prove directly that these forms are
canonical. Our proof provides explicit algorithms to compute all the blocks and
parameters in the canonical forms. We use these forms to derive canonical pairs
for simultaneous congruence of pairs of complex symmetric and skew-symmetric
matrices as well as canonical forms for simultaneous *congruence of pairs of
complex Hermitian matrices.Comment: 31 page
Representations of quivers and mixed graphs
This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman &
Hall/CRC, 2014. An informal introduction to representations of quivers and
finite dimensional algebras from a linear algebraist's point of view is given.
The notion of quiver representations is extended to representations of mixed
graphs, which permits one to study systems of linear mappings and bilinear or
sesquilinear forms. The problem of classifying such systems is reduced to the
problem of classifying systems of linear mappings
Superfluid to normal phase transition in strongly correlated bosons in two and three dimensions
Using quantum Monte Carlo simulations, we investigate the finite-temperature
phase diagram of hard-core bosons (XY model) in two- and three-dimensional
lattices. To determine the phase boundaries, we perform a finite-size-scaling
analysis of the condensate fraction and/or the superfluid stiffness. We then
discuss how these phase diagrams can be measured in experiments with trapped
ultracold gases, where the systems are inhomogeneous. For that, we introduce a
method based on the measurement of the zero-momentum occupation, which is
adequate for experiments dealing with both homogeneous and trapped systems, and
compare it with previously proposed approaches.Comment: 13 pages, 11 figures.
http://link.aps.org/doi/10.1103/PhysRevA.86.04362
Canonical matrices of bilinear and sesquilinear forms
Canonical matrices are given for
(a) bilinear forms over an algebraically closed or real closed field;
(b) sesquilinear forms over an algebraically closed field and over real
quaternions with any nonidentity involution; and
(c) sesquilinear forms over a field F of characteristic different from 2 with
involution (possibly, the identity) up to classification of Hermitian forms
over finite extensions of F.
A method for reducing the problem of classifying systems of forms and linear
mappings to the problem of classifying systems of linear mappings is used to
construct the canonical matrices. This method has its origins in representation
theory and was devised in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501].Comment: 44 pages; misprints corrected; accepted for publication in Linear
Algebra and its Applications (2007
Developing the Deutsch-Hayden approach to quantum mechanics
The formalism of Deutsch and Hayden is a useful tool for describing quantum
mechanics explicitly as local and unitary, and therefore quantum information
theory as concerning a "flow" of information between systems. In this paper we
show that these physical descriptions of flow are unique, and develop the
approach further to include the measurement interaction and mixed states. We
then give an analysis of entanglement swapping in this approach, showing that
it does not in fact contain non-local effects or some form of superluminal
signalling.Comment: 14 pages. Added section on entanglement swappin
A canonical form for nonderogatory matrices under unitary similarity
A square matrix is nonderogatory if its Jordan blocks have distinct
eigenvalues. We give canonical forms (i) for nonderogatory complex matrices up
to unitary similarity and (ii) for pairs of complex matrices up to similarity,
in which one matrix has distinct eigenvalues. The types of these canonical
forms are given by undirected and, respectively, directed graphs with no
undirected cycles.Comment: 18 page
Solution to the Equations of the Moment Expansions
We develop a formula for matching a Taylor series about the origin and an
asymptotic exponential expansion for large values of the coordinate. We test it
on the expansion of the generating functions for the moments and connected
moments of the Hamiltonian operator. In the former case the formula produces
the energies and overlaps for the Rayleigh-Ritz method in the Krylov space. We
choose the harmonic oscillator and a strongly anharmonic oscillator as
illustrative examples for numerical test. Our results reveal some features of
the connected-moments expansion that were overlooked in earlier studies and
applications of the approach
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