Eigenvalue Decomposition of a Parahermitian Matrix: Extraction of Analytic Eigenvectors

Weiss, Stephan; Proudler, Ian K.; Coutts, Fraser K.; Khattak, Faizan A.

doi:10.1109/tsp.2023.3269664

Cited by 16 publications

(3 citation statements)

References 52 publications

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“…The use of a trimmed sampling algorithm applied on the eigenvalues is proposed in [156] to replace the iterated eigenvalues for localization problems of large quantum systems. In [157], an iterative algorithm is proposed for the extraction of analytic eigen-vectors for decomposition of parahermitian matrices arising in broadband multiple-input multiple-output systems or array processing.…”

Section: Discussionmentioning

confidence: 99%

Eigenproblem Basics and Algorithms

Jäntschi

2023

Symmetry

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Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.

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Section: Discussionmentioning

confidence: 99%

Eigenproblem Basics and Algorithms

Jäntschi

2023

Symmetry

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“…The importance of the supply chain model in small and medium enterprises has been examined whose results are the discovery of the configuration pattern of the supply chain system of ATBM woven glove companies and the design of software for supply chain mapping (Weiss et al, 2023). The results of Soekesi's (2010) research related to the supply chain model are a comprehensive supply chain model that illustrates the formation of synergistic cooperation in the supply chain.…”

Section: Business Plan Designmentioning

confidence: 99%

Models of supply chain management in the efforts to develop the banyumasan batik industry

Susanti,

Nurlaili,

Darmawan

2023

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Inventory of raw materials is one of the crucial problems in the company. Errors in determining the size of raw material inventory is very influential for the smooth production of the company. PT Semen Bosowa Maros is a private company that focuses on cement production. This study aims to avoid excess and shortage of raw materials by calculating the amount of raw material inventory at PT Semen Bosowa Maros. This study uses the Min-Max Stock method. The research data was obtained from the company's annual data with related parties. The results of the study obtained safety stock values of Silica Sand 166.46 ton’s, Gypsum 1482.96 ton’s, Fly Ash 33.71 ton’s, and Trass 30.76 ton’s. Then, the minimum inventory calculation is carried out, it is known that the minimum inventory results are found in Trass raw materials of 260.18 ton’s while the results for maximum inventory are in Gypsum of 22117.18 ton’s. After calculating the min-max, an order policy can be obtained for each material, Silica Sand 1074.85 ton’s, Gypsum 9575.63 ton’s, Fly Ash 217.68 ton’s, and Trass 198.65 ton’s.

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“…An alternative approach uses polynomial matrices, which can simultaneously capture the correlations in space, time and frequency and is, therefore, appropriate for modelling multichannel broadband signals [9]. The processing of polynomial matrices has motivated the development of polynomial matrix eigenvalue decomposition (PEVD) algorithms in the z-domain based on the second-order sequential best rotation (SBR2) [10] and sequential matrix diagonalization (SMD) [11], [12], and those in the discrete Fourier transform (DFT)-domain [13], [14]. Unlike the KLT, the PEVD can mutually decorrelate signals for all lags [11]; this strong decorrelation together with spectral majorization of the decorrelated sequences guarantees optimality in the coding gain sense [2] Thus, the PEVD presents a solution for a finite order PCFB.…”

Section: Introductionmentioning

confidence: 99%

Signal Compaction Using Polynomial EVD for Spherical Array Processing With Applications

Neo,

Evers,

Weiss

et al. 2023

IEEE/ACM Trans. Audio Speech Lang. Process.

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Multi-channel signals captured by spatially separated sensors often contain a high level of data redundancy. A compact signal representation enables more efficient storage and processing, which has been exploited for data compression, noise reduction, and speech and image coding. This paper focuses on the compact representation of speech signals acquired by spherical microphone arrays. A polynomial matrix eigenvalue decomposition (PEVD) can spatially decorrelate signals over a range of time lags and is known to achieve optimum multichannel data compaction. However, the complexity of PEVD algorithms scales at best cubically with the number of channel signals, e.g., the number of microphones comprised in a spherical array used for processing. In contrast, the spherical harmonic transform (SHT) provides a compact spatial representation of the 3-dimensional sound field measured by spherical microphone arrays, referred to as eigenbeam signals, at a cost that rises only quadratically with the number of microphones. Yet, the SHT's spatially orthogonal basis functions cannot completely decorrelate sound field components over a range of time lags. In this work, we propose to exploit the compact representation offered by the SHT to reduce the number of channels used for subsequent PEVD processing. In the proposed framework for signal representation, we show that the diagonality factor improves by up to 7 dB over the microphone signal representation with a significantly lower computation cost. Moreover, when applying this framework to speech enhancement and source separation, the proposed method improves metrics known as short-time objective intelligibility (STOI) and source-to-distortion ratio (SDR) by up to 0.2 and 20 dB, respectively.

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Eigenvalue Decomposition of a Parahermitian Matrix: Extraction of Analytic Eigenvectors

Cited by 16 publications

References 52 publications

Eigenproblem Basics and Algorithms

Eigenproblem Basics and Algorithms

Models of supply chain management in the efforts to develop the banyumasan batik industry

Signal Compaction Using Polynomial EVD for Spherical Array Processing With Applications

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