A New Algorithm for Computing the Square Root of a Matrix

There are several different methods for computing a square root of a matrix. Previous research has been focused on Newton\u27s method and improving its speed and stability by application of Schur decomposition called Schur-Newton. In this thesis, we propose a new method for finding a square root of a matrix called the exponential method. The exponential method is an iterative method based on the matrix equation (X - I)^(2) = C, for C an n x n matrix, that finds an inverse matrix at the final step as opposed to every step like Newton\u27s method. We set up the matrix equation to form a 2n x 2n companion block matrix and then select the initial matrix C as a seed. With the seed, we run the power method for a given number of iterations to obtain a 2n x n matrix whose top block multiplied by the inverse of the bottom block is C^(1/2) + I. We will use techniques in linear algebra to prove that the exponential method converges to a specific square root of a matrix when it converges while numerical analysis techniques will show the rate of convergence. We will compare the outcomes of the exponential method versus Schur-Newton, and discuss further research and modifications to improve its versatility

More on N=1 Matrix Model Curve for Arbitrary N

Using both the matrix model prescription and the strong-coupling approach, we describe the intersections of n=0 and n=1 non-degenerated branches for quartic (polynomial of adjoint matter) tree-level superpotential in N=1 supersymmetric SO(N)/USp(2N) gauge theories with massless flavors. We also apply the method to the degenerated branch. The general matrix model curve on the two cases we obtain is valid for arbitrary N and extends the previous work from strong-coupling approach. For SO(N) gauge theory with equal massive flavors, we also obtain the matrix model curve on the degenerated branch for arbitrary N. Finally we discuss on the intersections of n=0 and n=1 non-degenerated branches for equal massive flavors.Comment: 36pp; to appear in JHE

Random matrix theory and symmetric spaces

In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important results can be obtained from this identification. In particular the Cartan classification of triplets of symmetric spaces with positive, zero and negative curvature gives rise to a new classification of random matrix ensembles. The review is organized into two main parts. In Part I the theory of symmetric spaces is reviewed with particular emphasis on the ideas relevant for appreciating the correspondence with random matrix theories. In Part II we discuss various applications of symmetric spaces to random matrix theories and in particular the new classification of disordered systems derived from the classification of symmetric spaces. We also review how the mapping from integrable Calogero--Sutherland models to symmetric spaces can be used in the theory of random matrices, with particular consequences for quantum transport problems. We conclude indicating some interesting new directions of research based on these identifications.Comment: 161 pages, LaTeX, no figures. Revised version with major additions in the second part of the review. Version accepted for publication on Physics Report

Calogero-Sutherland techniques in the physics of disorderd wires

We discuss the connection between the random matrix approach to disordered wires and the Calogero-Sutherland models. We show that different choices of random matrix ensembles correspond to different classes of CS models. In particular, the standard transfer matrix ensembles correspond to CS model with sinh-type interaction, constructed according to the $C_n$ root lattice pattern. By exploiting this relation, and by using some known properties of the zonal spherical functions on symmetric spaces we can obtain several properties of the Dorokhov-Mello-Pereyra-Kumar equation, which describes the evolution of an ensemble of quasi one-dimensional disordered wires of increasing length $L$. These results are in complete agreement with all known properties of disordered wires. (To appear in the Proceedings of the Conference: Recent Developments in Statistical Mechanics and Quantum Field Theory (Trieste, 1995))Comment: 16 pages, Late

Quantum superalgebras at roots of unity and non-abelian symmetries of integrable models

We consider integrable vertex models whose Boltzmann weights (R-matrices) are trigonometric solutions to the graded Yang-Baxter equation. As is well known the latter can be generically constructed from quantum affine superalgebras $U_{q}(\hat g)$. These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter $q$ when periodic boundary conditions are imposed. If $q$ is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non-abelian subalgebras which are translation invariant and supercommute with the transfer matrix and therefore with all charges of the model. In the line of argument we introduce the restricted quantum superalgebra $U^{res}_q(\hat g)$ and investigate its root of unity limit. We prove several new formulas involving supercommutators of arbitrary powers of the Chevalley-Serre generators and derive higher order quantum Serre relations as well as an analogue of Lustzig's quantum Frobenius theorem for superalgebras.Comment: 31 pages, tcilatex (minor typos corrected

Auxiliary matrices on both sides of the equator

The spectra of previously constructed auxiliary matrices for the six-vertex model at roots of unity are investigated for spin-chains of even and odd length. The two cases show remarkable differences. In particular, it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to Baxter's TQ-equation corresponding to the Bethe ansatz equations above and below the equator. In contrast, one finds for even spin-chains only one linear independent solution and complete strings. The other main result is the proof of a previous conjecture on the degeneracies of the six-vertex model at roots of unity. The proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight-vertex model conjectured by Fabricius and McCoy.Comment: 22 pages; 2nd version: one paragraph added in the conclusion and some typos correcte

Square Root Singularity in Boundary Reflection Matrix

Two-particle scattering amplitudes for integrable relativistic quantum field theory in 1+1 dimensions can normally have at most singularities of poles and zeros along the imaginary axis in the complex rapidity plane. It has been supposed that single particle amplitudes of the exact boundary reflection matrix exhibit the same structure. In this paper, single particle amplitudes of the exact boundary reflection matrix corresponding to the Neumann boundary condition for affine Toda field theory associated with twisted affine algebras $a_{2n}^{(2)}$ are conjectured, based on one-loop result, as having a new kind of square root singularity.Comment: 10 pages, latex fil