Search results for "Matrix"
showing 10 items of 3205 documents
Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables
2014
28 pages; International audience; We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to…
Duality of reduced density matrices and their eigenvalues
2014
For states of quantum systems of N particles with harmonic interactions we prove that each reduced density matrix ρ obeys a duality condition. This condition implies duality relations for the eigenvalues λk of ρ and relates a harmonic model with length scales ${{\ell }_{1}},{{\ell }_{2}},\ldots ,{{\ell }_{N}}$ with another one with inverse lengths $1/{{\ell }_{1}},1/{{\ell }_{2}},\ldots ,1/{{\ell }_{N}}$. Entanglement entropies and correlation functions inherit duality from ρ. Self-duality can only occur for noninteracting particles in an isotropic harmonic trap.
Affine equivariant multivariate rank methods
2003
The classical multivariate statistical methods (MANOVA, principal component analysis, multivariate multiple regression, canonical correlation, factor analysis, etc.) assume that the data come from a multivariate normal distribution and the derivations are based on the sample covariance matrix. The conventional sample covariance matrix and consequently the standard multivariate techniques based on it are, however, highly sensitive to outlying observations. In the paper a new, more robust and highly efficient, approach based on an affine equivariant rank covariance matrix is proposed and outlined. Affine equivariant multivariate rank concept is based on the multivariate Oja (Statist. Probab. …
Non-self-adjoint Hamiltonians with complex eigenvalues
2016
Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.
Quadratic ${\mathcal P}{\mathcal T}$-symmetric operators with real spectrum and similarity to self-adjoint operators
2012
It is established that a -symmetric elliptic quadratic differential operator with real spectrum is similar to a self-adjoint operator precisely when the associated fundamental matrix has no Jordan blocks.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
On the usage of joint diagonalization in multivariate statistics
2022
Scatter matrices generalize the covariance matrix and are useful in many multivariate data analysis methods, including well-known principal component analysis (PCA), which is based on the diagonalization of the covariance matrix. The simultaneous diagonalization of two or more scatter matrices goes beyond PCA and is used more and more often. In this paper, we offer an overview of many methods that are based on a joint diagonalization. These methods range from the unsupervised context with invariant coordinate selection and blind source separation, which includes independent component analysis, to the supervised context with discriminant analysis and sliced inverse regression. They also enco…
Standard forms and entanglement engineering of multimode Gaussian states under local operations
2007
We investigate the action of local unitary operations on multimode (pure or mixed) Gaussian states and single out the minimal number of locally invariant parametres which completely characterise the covariance matrix of such states. For pure Gaussian states, central resources for continuous-variable quantum information, we investigate separately the parametre reduction due to the additional constraint of global purity, and the one following by the local-unitary freedom. Counting arguments and insights from the phase-space Schmidt decomposition and in general from the framework of symplectic analysis, accompany our description of the standard form of pure n-mode Gaussian states. In particula…
Separation of Uncorrelated Stationary time series using Autocovariance Matrices
2015
Blind source separation (BSS) is a signal processing tool, which is widely used in various fields. Examples include biomedical signal separation, brain imaging and economic time series applications. In BSS, one assumes that the observed $p$ time series are linear combinations of $p$ latent uncorrelated weakly stationary time series. The aim is then to find an estimate for an unmixing matrix, which transforms the observed time series back to uncorrelated latent time series. In SOBI (Second Order Blind Identification) joint diagonalization of the covariance matrix and autocovariance matrices with several lags is used to estimate the unmixing matrix. The rows of an unmixing matrix can be deriv…
The Induced Smoothed lasso: A practical framework for hypothesis testing in high dimensional regression.
2020
This paper focuses on hypothesis testing in lasso regression, when one is interested in judging statistical significance for the regression coefficients in the regression equation involving a lot of covariates. To get reliable p-values, we propose a new lasso-type estimator relying on the idea of induced smoothing which allows to obtain appropriate covariance matrix and Wald statistic relatively easily. Some simulation experiments reveal that our approach exhibits good performance when contrasted with the recent inferential tools in the lasso framework. Two real data analyses are presented to illustrate the proposed framework in practice.
Tests and estimates of shape based on spatial signs and ranks
2009
Nonparametric procedures for testing and estimation of the shape matrix in the case of multivariate elliptic distribution are considered. Testing for sphericity is an important special case. The tests and estimates are based on the spatial sign and rank covariance matrices. The estimates based on the spatial sign covariance matrix and symmetrized spatial sign covariance matrix are Tyler's [A distribution-free M-estimator of multivariate scatter, Ann. Statist. 15 (1987), pp. 234–251] shape matrix and and Dümbgen's [On Tyler's M-functional of scatter in high dimension, Ann. Inst. Statist. Math. 50 (1998), pp. 471–491] shape matrix, respectively. The test based on the spatial sign covariance m…