Eigendecomposition Of Matrix Research Articles

Characterization of subsurface defects in aeronautical riveted lap-joints using multi-frequency eddy current imaging

The authors present an original eddy current imager (ECI) designed for the fast and accurate non-destructive evaluation of defects buried next to rivets in aeronautical lap-joints. The ECI is associated to a signal processing method based on a principal component analysis (PCA) followed by a maximum likelihood (ML) approach. The PCA was implemented using EC images obtained with selected excitation frequencies. These images are considered as resulting from a linear mixing of different sources including the presence of rivets and defects, and the PCA is used to separate these sources thanks to an eigen decomposition of the EC data covariance matrix. As a result, the defect signatures are enhanced and used to implement an automatic defect characterization. This characterization is carried out by the means of an ML approach which allows the length and depth of the defects to be estimated. The method was implemented for the evaluation of a laboratory made riveted lap joint mock-up featuring buried defects. It was experimentally optimized and successfully implemented for the characterization of calibrated defects ranging from 2 to 10 mm in length and 2 to 8 mm in depth.

Discrete fractional Fourier transform based on the eigenvectors of tridiagonal and nearly tridiagonal matrices

The recent emergence of the discrete fractional Fourier transform (DFRFT) has caused a revived interest in the eigenanalysis of the discrete Fourier transform (DFT) matrix F with the objective of generating orthonormal Hermite-Gaussian-like eigenvectors. The Grunbaum tridiagonal matrix T-which commutes with matrix F-has only one repeated eigenvalue with multiplicity two and simple remaining eigenvalues. A detailed eigendecomposition of matrix T is performed with the objective of deriving two orthonormal eigenvectors-common to both the F and T matrices-pertaining to the repeated eigenvalue of T. The nearly tridiagonal matrix S first introduced by Dickinson and Steiglitz and later studied by Candan et al.-which commutes with matrix F-is rigorously proved to reduce to a 2x2 block diagonal form by means of a similarity transformation defined in terms of an involutary matrix P. Moreover explicit expressions are derived for the elements of the two tridiagonal submatrices forming the two diagonal blocks in order to circumvent the need for performing two matrix multiplications. Although matrix T has the merit of being tridiagonal and does not need the tridiagonalization step as matrix S, the simulation results show that the eigenvectors of matrix S better approximate samples of the Hermite-Gaussian functions than those of matrix T and moreover they have a shorter computation time due to the block diagonalization result. Consequently they can serve as a better basis for developing the DFRFT.

Normalized proper orthogonal decomposition (NPOD) for building pressure data compression

Proper orthogonal decomposition (POD) has been widely realized as a tool for compressing fluctuating building pressure data in wind-engineering area. Modes are determined by eigen decomposition of data covariance matrix, and then truncation is applied to retain only the modes with the highest energy. However, as observed by S. Kho, C. Baker, R. Hoxey [Pod/arma reconstruction of the surface pressure field around a low rise structure, J. Wind Eng. Ind. Aerodyn. (90) (2002) 1831–1842] and analyzed in the authors’ previous paper D. Ruan, H. He, D. Smith, K.C. Mehta [A Semi-optimal Mode Selection Scheme for Pod Based Compression of Wind Field Data, Seoul, Korea, 2004], reconstruction performance varies a lot among individual taps. In D. Ruan, H. He, D. Smith, K.C. Mehta [A Semi-optimal Mode Selection Scheme for Pod Based Compression of Wind Field Data, Seoul, Korea, 2004], a semi-optimal mode choosing scheme was proposed in an integer programming (IP) framework and numerical experiments have shown sound results. Nevertheless, computation cost for solving IP optimization could be an issue as the total number of taps grows. In this paper, we address the problem with normalized proper orthogonal decomposition which is from a completely different perspective: we first normalize the data to force every tap to have similar contribution to the total energy; the standard POD is then applied to get a compact representation. During the reconstruction phase, normalized data is first restored and a scaling procedure which is exactly the inverse to the normalization is applied to finally reconstitute the data in its original form. Feasibility test with experimental data from Texas Tech University yields good results. Computation cost is almost the same as the standard POD method.

AHP에서 왜대칭행렬의 고유분해를 이용한 중요도 추정법의 제안

AHP기법에서는 의사결정 요소들의 중요도를 추정함에 있어 통상 쌍대비교행렬 그 자체에 고유벡터법 또는 대수최소제곱법을 적용한다. 본 연구에서는 왜대칭행렬의 고유분해를 통해 쌍대비교행렬을 조정한 후 조정된 쌍대비교행렬에 대해 고유벡터법 또는 대수최소제곱법을 적용하는 중요도 추정법을 제안한다. 그리고 이 추정법이 가지는 여러 가지 이점과 의미를 이론적 근거와 실제 사용 예를 통해 보이고자 한다. 본 연구결과는 불일치성이 높은 쌍대비교행렬이 주어진 경우 불일치성을 줄이는데 특히 유용하게 활용될 수 있을 것이다. Generally to estimate the priority vector in AHP, an eigen-vector method or a log-arithmic least square method is applied to pairwise comparison matrix itself. In this paper an estimating method is suggested, which is applied to pairwise comparison matrix adjusted by using the eigen-decomposition of skew-symmetric matrix. We also show theoretical background, meanings, and several advantages of this method by example. This method may be useful in case that pairwise comparison matrix is quite inconsistent.

Direction finding in partly calibrated sensor arrays composed of multiple subarrays

We consider the direction-finding problem in partly calibrated arrays composed of several calibrated and identically oriented (but possibly nonidentical) subarrays that are displaced by unknown (and possibly time-varying) vector translations. A new search-free eigenstructure-based direction-finding approach is proposed for such class of sensor arrays. It is referred to as the rank-reduction (RARE) estimator and enjoys simple implementation that entails computing the eigendecomposition of the sample array covariance matrix and polynomial rooting. Closed-form expressions for the deterministic Cramer-Rao bounds (CRBs) on direction-of-arrival (DOA) estimation for the considered class of sensor arrays are derived. Comparison of these expressions with simulation results show that the finite-sample performance of RARE algorithms in both time-invariant and time-varying array cases is close to the corresponding bounds. Moreover, comparisons of the derived CRBs with the well-known bounds for the fully calibrated time-invariant array case help to discover several interesting properties of DOA estimation in partly calibrated and time-varying arrays.

A new discrete fractional Fourier transform based on constrained eigendecomposition of DFT matrix by Lagrange multiplier method

This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier transform and by performing a novel error-removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, several numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT.

Parallel implementation of linear algebra problems on Dawning-1000

In this paper, some parallel algorithms are described for solving numerical linear algebra problems on Dwning-1000. They include matrix multiplication,LU factorization of a dense matrix, Cholesky factorization of a symmetric matrix, and eigendecomposition of symmetric matrix for real and complex data types. These programs are constructed based on fast BLAS library of Dawning-1000 under NX environment. Some comparison results under different parallel environments and implementing methods are also given for Cholesky factorization. The execution time, measured performance and speedup for each problem on Dawning-1000 are shown. For matrix multiplication andLU factorization, 1.86GFLOPS and 1.53GFLOPS are reached.

A robust adaptive array based on signal subspace approach

In this note, we propose a signal subspace approach that improves the performance of a beam steered adaptive array in the presence of steering errors due to look-direction error (LDE) and/or random steering error (RSE). In the method, the degrees of freedom (DOF) are reduced so as not to cancel the desired signal while preserving the optimal characteristic of the array, and thus the weights of the array are determined by a linear combination of the eigenvectors of the signal subspace. The proposed method works as far as the eigen decomposition of the input covariance matrix into signal and noise subspaces is possible. The proposed method improves noticeably the array performance of the beam steered array in the presence of steering errors and provides the optimum array performance in the absence of steering errors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Two-dimensional coherent noise suppression in seismic data using eigendecomposition

A method for the suppression of coherent noise in seismic data based on the eigendecomposition of a data covariance matrix is demonstrated. Based on the Karhunen-Loeve transform, the proposed procedure is useful against noise energy exhibiting both two-dimensional space and time coherencies or coherent two-dimensional patterns which are not necessarily linear and therefore cannot generally be velocity-filtered. This method trains on a region containing the undesired coherent noise; the dominant eigenvectors determined from the covariance matrix of that noise are used to reconstruct the noise in the region of interest. Subtracting the reconstruction from the original data leaves a residual in which the coherent noise has been suppressed. In the example considered, this method effectively suppresses the noise in a record of marine seismic data containing backscattered source energy. >