Complex Hermitian Matrix Research Articles

Direct Adaption Methods for Linear and Circular Antenna Arrays

Introduction. The mitigation of interferences that degrade the performance of navigation systems constitutes one of the most significant problems of contemporary satellite navigation. This problem is conventionally solved using digital adaptive space filters. Depending on a particular radio technical system, the mathematical description of digital signal processing methods may involve specific calculation structures implemented using specific calculation algorithms. For example, the use of centrosymmetric linear and circular antenna arrays in a radio navigation system allows the description of such systems in terms of Toeplitz and circulant sample covariance matrices, respectively, and the inversion of such matrices by means of special numerical methods in order to design a digital filter.Aim. A comparative analysis of the performance of space signal processing algorithms is carried out along with an estimation of Toeplitz and circulant sample covariance matrices and numerical methods of their inversion. The previously obtained results in this field are clarified.Materials and methods. An analysis of algorithm performance was carried out in the MATLAB environment using experimental recordings of satellite navigation signals and jammers obtained by an actual radio technical system.Results. A new expression was derived for estimating circulant sample covariance matrices. Formulae that describe a modification of the Bareiss numerical Toeplitz matrix inversion algorithm for the case of complex Hermitian matrix were introduced. An analysis of the results of computer simulation allowed the algorithms with the highest performance to be indicated. The amount of time taken by the algorithms based on Toeplitz and circulant matrices did not exceed 2.5 10⋅ −3 s and 0.04 s, respectively. The carrier-to-noise ratio in the processed signal was at least 46 dB. Conclusion. The formulae obtained and the algorithms analyzed can be used when implementing adaptive digital filtering of satellite navigation signals.

Efficient parallel reduction of bandwidth for symmetric matrices

Bandwidth reduction can be a first step in the computation of eigenvalues and eigenvectors for a wide-banded complex Hermitian (or real symmetric) matrix. We present algorithms for this reduction and the corresponding back-transformation of the eigenvectors. These algorithms rely on blocked Householder transformations, thus enabling level 3 BLAS performance, and they feature two levels of parallelism. The efficiency of our approach is demonstrated with numerical experiments.

Birkhoff-James orthogonality in the trace norm, with applications to quantum resource theories

Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.

The spectral spread of Hermitian matrices

Let A be an n×n complex Hermitian matrix and let λ(A)=(λ1,…,λn)∈Rn denote the eigenvalues of A, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of A, denoted Spr+(A), given by Spr+(A)=(λ1−λn,λ2−λn−1,…,λk−λn−k+1)∈Rk, where k=[n/2] (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of A, that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.

On C $$\varvec{\otimes }$$ H $$\varvec{\otimes }$$ O-Valued Gravity, Sedenions, Hermitian Matrix Geometry and Nonsymmetric Kaluza–Klein Theory

We review briefly how R $$\otimes $$ C $$\otimes $$ H $$\otimes $$ O-valued gravity (real-complex-quaterno-octonionic gravity) naturally can describe a grand unified field theory of Einstein’s gravity with a Yang–Mills theory containing the Standard Model group $$SU(3) \times SU(2) \times U(1)$$ . The algebra of left actions associated with the composite algebras involving the Division algebras, and the Sedenions $$\mathbf{S}$$ , and acting on themselves, all lead to complex Clifford algebras (complex matrix algebras). The complex Cl(16) algebra is the most appealing one since it is the one corresponding to the algebra of left actions of $$\mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O} \otimes \mathbf{S}$$ acting on itself, and containing the $$\mathbf{e_8 \oplus e_8}$$ algebra of the anomaly free 10D Heterotic String. An analysis of $$ C \otimes H \otimes O$$ -valued gravity reveals that it bears a connection to Nonsymmetric Kaluza–Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $$ C \otimes H \otimes O$$ -valued metrics in two complex dimensions with complex metrics in higher dimensional real manifolds ( $$ D = 32 $$ real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.

Definitive Conditions on Maxwell's Tangential E or H for Solutions to His Equations Inside a Bounded Region

AbstractLet U be a connected, closed, bounded region in ℝ3 with smooth boundary 𝛛U. Consider Maxwell's equations on U for smooth fields and smooth sources, which are time harmonic, i.e., the fields E,B,H,D and the current density J and charge density ρ on U temporally varying as e−iωt. We assume ω ≠ 0; when it is real, ω is the frequency of the pure oscillation; when ω is not real, one has exponentially decreasing or increasing oscillatory fields. We treat very general electric permittivity ɛω and magnetic permeability μω, which may both depend upon ω. Both can be arbitrary, time‐independent, smoothly varying with position and described by complex Hermitian tensors subject only to weak positivity conditions specified below at each point of U.In this paper necessary and sufficient conditions are found for the tangential components, ET or HT on 𝛛U, to be realized by solutions of Maxwell's equations on all of U with these given sources J, ρ in the time‐harmonic realm with ω ≠ 0. Also, all solutions E,H on U with specified ET, HT, respectively, satisfying the necessary finite‐dimensional conditions are derived by an effective natural algorithm.In the electric ET case, the positivity condition is this: the complex Hermitian matrix μω(p) is to be positive definite while only the real part of ɛω(p), i.e., Re(ɛω(p)), necessarily real symmetric, need be positive definite. In the magnetic‐type HT case, the two roles are reversed. These positivity conditions are more general than those commonly considered. In the final section Maxwell's equations and all the present results are reformulated yet more generally in a context of compact, smooth, Riemannian manifolds with boundary. Sample examples include periodic lattices and wave guides. © 2019 Wiley Periodicals, Inc.

On the power method for quaternion right eigenvalue problem

In this paper, we study the power method of the right eigenvalue problem of a quaternion matrix A . If A is Hermitian, we propose the power method that is a direct generalization of that of complex Hermitian matrix. When A is non-Hermitian, by applying the properties of quaternion right eigenvalues, we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector. We also briefly discuss the inverse power method and shift inverse power method for the both cases. The real structure-preserving algorithm of the power method in the two cases are also proposed, and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method.

A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a thick-restart version of the Lanczos algorithm with deflation (``locking'') and a new type of polynomial filter obtained from a least-squares technique. The resulting algorithm can be utilized in a “spectrum-slicing” approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different subintervals independently from one another.

Numerical determination for solving the symmetric eigenvector problem using genetic algorithm

The eigenvalues and eigenvectors of a matrix have many applications in engineering and science. For example they are important in studying and solving structural problems, in the treatment of signal or image processing, in the study of quantum mechanics and in certain physical problems. It is therefore essential to analyze methodologies to obtain the eigenvectors and eigenvalues ​​of symmetric and Hermitian matrices. In this paper the authors present a methodology for obtaining the eigenvectors and eigenvalues of a symmetric or Hermitian matrix using a genetic algorithm. Unlike other methodologies, the process is centred in searching the eigenvectors and calculating the eigenvalues afterwards. In the search of the eigenvectors a genetic-based algorithm is used. Genetic algorithms are indicated when the search space is extended, unknown or with an intricate geometry. Also, the target vector space can be either real or complex, allowing in this way a wider field of application for the proposed method. The algorithm is tested comparing the results with those obtained by other methods or with the values previously known. So, seven applications are included: a real symmetric matrix corresponding to a vibrating system, a complex Hermitian matrix and an important application of the diagonalization problem (Coope matrix) corresponding to quantum mechanics examples, a physical problem in which data are analysed to reduce the number of variables, a comparison with the power method and the studies of a degenerate and an ill-conditioned matrix.

A Matricial Proof of the Symmetric Exchange Axiom for Eigenvalues of Principal Submatrices of a Complex Hermitian Matrix

In [C.R. Johnson, B. Kroschel, M. Omladič, Eigenvalue multiplicities in principal submatrices, Linear Algebra Appl. 390 (2004) 111-120] a result constraining the eigenvalues of principal submatrices of complex Hermitian matrices, based upon matroid theory, was given. Here we give a matricial proof of this result which also enables us to find a generalization of the original result.

Real Fast Structure-Preserving Algorithm for Eigenproblem of Complex Hermitian Matrices

It is well known that the flops for complex operations are usually 4 times of real cases. In the paper, using real operations instead of complex, a real fast structure-preserving algorithm for eigenproblem of complex Hermitian matrices is given. We make use of the real symmetric and skew-Hamiltonian structure transformed by Wilkinson's way, focus on symplectic orthogonal similarity transformations and their structure-preserving property, and then reduce it into a two-by-two block tridiagonal symmetric matrix. Finally a real algorithm can be quickly obtained for eigenvalue problems of the original Hermitian matrix. Numerical experiments show that the fast algorithm can solve real complex Hermitian matrix efficiently, stably, and with high precision.

2 × 2 random matrix ensembles with reduced symmetry: from Hermitian to -symmetric matrices

A possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional Hermitian matrices or parity–time ()-symmetric matrices. To illustrate the main idea, we first study 2 × 2 complex Hermitian matrix ensembles with O(2)-invariant constraints, yielding novel level-spacing statistics such as singular distributions, the half-Gaussian distribution, distributions interpolating between the GOE (Gaussian orthogonal ensemble) distribution and half-Gaussian distributions, as well as the gapped-GOE distribution. Such a symmetry-reduction strategy is then used to explore 2 × 2 -symmetric matrix ensembles with real eigenvalues. In particular, -symmetric random matrix ensembles with U(2) invariance can be constructed, with the conventional complex Hermitian random matrix ensemble being a special case. In two examples of -symmetric random matrix ensembles, the level-spacing distributions are found to be the standard GUE (Gaussian unitary ensemble) statistics or the ‘truncated-GUE’ statistics.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

When combined with Krylov projection methods, polynomial filtering can provide a powerful method for extracting extreme or interior eigenvalues of large sparse matrices. This general approach can be quite efficient in the situation when a large number of eigenvalues is sought. However, its competitiveness depends critically on a good implementation. This paper presents a technique based on such a combination to compute a group of extreme or interior eigenvalues of a real symmetric (or complex Hermitian) matrix. The technique harnesses the effectiveness of the Lanczos algorithm with partial reorthogonalization and the power of polynomial filtering. Numerical experiments indicate that the method can be far superior to competing algorithms when a large number of eigenvalues and eigenvectors is to be computed.

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.

Calculation of transmission probability by solving an eigenvalue problem

The electron transmission probability in nanodevices is calculated by solving an eigenvalueproblem. The eigenvalues are the transmission probabilities and the number of nonzeroeigenvalues is equal to the number of open quantum transmission eigenchannels. Thenumber of open eigenchannels is typically a few dozen at most, thus the computational costamounts to the calculation of a few outer eigenvalues of a complex Hermitian matrix(the transmission matrix). The method is implemented on a real space grid basisproviding an alternative to localized atomic orbital based quantum transportcalculations. Numerical examples are presented to illustrate the efficiency of the method.

Distinct Modes of the East Asian Summer Monsoon*

Abstract Resolution of a complex Hermitian matrix derived from monthly mean 850-hPa wind fields during the summer season (June–August) from 1968 to 2004 revealed four different modes of East Asian summer monsoon (EASM) variability. The leading EASM mode, accounting for 19.6% of the variance, is characterized by two different modes (M11 and M12) or their combination. Both portray a closed cyclonic or anticyclonic circulation anomaly over the western North Pacific (WNP), South China Sea (SCS), and southeastern China; corresponding anomalous geopotential height fields show a wave train structure from the WNP across Japan, the Okhotsk Sea, and Alaska to North America. Thus, the leading EASM mode characterizes the teleconnection pattern of the WNP-EASM. The correlation between M11 (M12) and the dynamic index for the WNP-EASM is 0.85 (0.51). M11 has leading spectral peaks at 15 and 3 yr, whereas M12 displays a predominant peak at 2 yr. It is found that M11 has interdecadal variations, with the transition years being circa 1973 and 1989, respectively. M11 is closely related to air–sea interactions in the SCS and the northwestern Pacific, and its association with the convective heat source over the northwestern Pacific is secondary. In contrast, M12 is closely related to the tropical convective heat source rather than tropical western Pacific sea surface temperature (SST). The second EASM mode, accounting for 12.8% of the variance, is identified and characterized by two distinct and alternating modes or their linear combination (M21 and M22). One mode (M21) closely relates to the dual blocking high pattern detected in anomalous sea level pressure (SLP) and 500-hPa geopotential heights over the Ural Mountains and the Okhotsk Sea. The other (M22) corresponds to a dipole blocking anomaly in anomalous SLP and geopotential heights, with opposing anomalous centers in the south of Japan and the Korean peninsula, and the area between Lake Baikal and the Okhotsk Sea. M22 shows significant correlations with summer mean rainfall in southern and southeastern China. Thus, a single index of EASM is inappropriate for investigating and predicting the EASM.

Stopping Criteria for Rational Matrix Functions of Hermitian and Symmetric Matrices

Building upon earlier work by Golub, Meurant, Strakoš, and Tichý, we derive new a posteriori error bounds for Krylov subspace approximations to $f(A)b$, the action of a function f of a real symmetric or complex Hermitian matrix A on a vector b. To this purpose we assume that a rational function in partial fraction expansion form is used to approximate f, and the Krylov subspace approximations are obtained as linear combinations of Galerkin approximations to the individual terms in the partial fraction expansion. Our error estimates come at very low computational cost. In certain important special situations they can be shown to actually be lower bounds of the error. Our numerical results include experiments with the matrix exponential, as used in exponential integrators, and with the matrix sign function, as used in lattice quantum chromodynamics simulations, and demonstrate the accuracy of the estimates. The use of our error estimates within acceleration procedures is also discussed.

Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices

This report may be considered as a non-trivial extension of an unpublished report by William Kahan ( Accurate Eigenvalues of a symmetric tri-diagonal matrix, Technical Report CS 41, Computer Science Department, Stanford University, 1966). His interplay between matrix theory and computer arithmetic led to the development of algorithms for computing accurate eigenvalues and singular values. His report is generally considered as the precursor for the development of IEEE standard 754 for binary arithmetic. This standard has been universally adopted by virtually all PC, workstation and midrange hardware manufactures and tens of billions of such machines have been produced. Now we use the features in this standard to improve the original algorithm. In this paper, we describe an algorithm in floating-point arithmetic to compute the exact inertia of a real symmetric (shifted) tridiagonal matrix. The inertia, denoted by the integer triplet ( π, ν, ζ), is defined as the number of positive, negative and zero eigenvalues of a real symmetric (or complex Hermitian) matrix and the adjective exact refers to the eigenvalues computed in exact arithmetic. This requires the floating-point computation of the diagonal matrix D of the LDL t factorization of the shifted tridiagonal matrix T − τI with +∞ and −∞ rounding modes defined in IEEE 754 standard. We are not aware of any other algorithm which gives the exact answer to a numerical problem when implemented in floating-point arithmetic in standard working precisions. The guaranteed intervals for eigenvalues are obtained by bisection or multisection with this exact inertia information. Similarly, using the Golub–Kahan form, guaranteed intervals for singular values of bidiagonal matrices can be computed. The diameter of the eigenvalue (singular value) intervals depends on the number of shifts with inconsistent inertia in two rounding modes. Our algorithm not only guarantees the accuracy of the solutions but is also consistent across different IEEE 754 standard compliant architectures. The unprecedented accuracy provided by our algorithms could be also used to debug and validate standard floating-point algorithms for computation of eigenvalues (singular values). Accurate eigenvalues (singular values) are also required by certain algorithms to compute accurate eigenvectors (singular vectors). We demonstrate the accuracy of our algorithms by using standard matrix examples. For the Wilkinson W 21 + matrix, the eigenvalues (in IEEE double precision) are very accurate with an (open) interval diameter of 6 ulps (units of the last place held of the mantissa) for one of the eigenvalues and lesser (down to 2 ulps) for others. These results are consistent across many architectures including Intel, AMD, SGI and DEC Alpha. However, by enabling IEEE double extended precision arithmetic in Intel/AMD 32-bit architectures at no extra computational cost, the (open) interval diameters were reduced to one ulp, which is the best possible solution for this problem. We have also computed the eigenvalues of a tridiagonal matrix which manifests in Gauss–Laguerre quadrature and the results are extremely good in double extended precision but less so in double precision. To demonstrate the accuracy of computed singular values, we have also computed the eigenvalues of the Kac 30 matrix, which is the Golub–Kahan form of a bidiagonal matrix. The tridiagonal matrix has known integer eigenvalues. The bidiagonal Cholesky factor of the Gauss–Laguerre tridiagonal is also included in the singular value study.

RESPONSE OF ENERGY LEVELS AND WAVEFUNCTIONS OF 2-D ARTIFICIAL ATOMS TO CHANGES IN PARAMETERS IN THE HAMILTONIAN

We explore the pattern of evolution of energy levels and wavefunctions in a harmonically confined single electron artificial atom in 2 dimensions, in the presence of a perpendicular magnetic field and anharmonic perturbations. A configuration interaction (CI) type of approach in a basis of appropriate harmonic oscillator eigenfunctions is used, leading to a complex sparse hermitian matrix eigenvalue problem. We show that for a given strength of the confining potential descriptors like quantum information entropy, level spacing distribution, degree of localization of one electron density and probability current point to a threshold value for the cyclotron frequency (ωc), beyond which there appears to be a transition from a delocalized description rather than a localized one. No signature of quantum chaos is detected.

Matrix representation of vector potential: DVR and TDDVR formulations and dynamics

The inclusion of the geometric phase effects through the addition of vector potential is well known. We present the formulation of DVR and TDDVR matrix equations for any 2-D system with vector potential. The effective potential appears as the complex hermitian matrix in the DVR/TDDVR representation where in case of TDDVR, matrices associated with “classical” momentum also plays an important role in the dynamics. We derive the rigorous expressions of “classical” equations of motion from Dirac–Frenkel variational principle without introducing the “classical” path as such. Numerical calculations by using DVR/TDDVR equations have been carried out to obtain the signature of geometric phase on the reactive and non-reactive scattering processes. TDDVR appears to be better compromise between speed and accuracy than traditional quantum dynamics numerical methodologies (DVR/FFT).