Symmetric Matrix Research Articles

Symmetric Double-Supplemented Nested Array for Passive Localization of Mixed Near-Field and Far-Field Sources

In mixed-field source localization, the physical properties of a sensor array, such as the degrees of freedom (DOFs), aperture, and coupling leakage, directly affect the accuracy of estimating the direction of arrival (DOA). Compared to conventional symmetric uniform linear arrays, symmetric non-uniform linear arrays (SNLAs) have a greater advantage in mixed-field source localization due to their larger aperture and higher DOF. However, current SNLAs require improvements in their physical properties through modifications to the array structure in order to achieve more accurate source localization estimates. Therefore, this study proposes a symmetric double-supplemented nested array (SDSNA), which translates nested subarrays based on symmetric nested arrays to increase the aperture and inserts two symmetric supplemented subarrays to fill the holes created by the translation. This method results in longer consecutive difference coarray lags and larger apertures. The SDSNA is compared to existing advanced SNLAs in terms of their physical properties and DOA estimation. The results show that, with the same number of sensors, the SDSNA has a higher DOF, a larger aperture, and smaller coupling, indicating the advantages of the SDSNA in terms of its physical properties. Under the same experimental conditions, the SDSNA has a lower root-mean-square error of source location, thus indicating better performance in terms of both DOA and distance estimation.

Lattice factorization based causal symmetric paraunitary matrix extension and construction of symmetric orthogonal multiwavelets

In this paper, we propose a lattice factorization based symmetric paraunitary matrix extension method to design a causal symmetric paraunitary multifilter banks(PUMFBs) and construct compactly supported symmetric orthogonal multiwavelets by using the method.First, we transform a Laurent polynomial matrix consisting of polyphase components of a given symmetric orthogonal multifilter into a Laurent polynomial matrix with a simple structure, and then obtain a lattice factorization based symmetric paraunitary extension matrix by factorizing the transformed Laurent polynomial matrix and extending the constant matrix of the factorization. We also construct symmetric orthogonal multiwavelets by the proposed symmetric paraunitary matrix extension method. Finally, some examples are given to illustrate the proposed symmetric paraunitary matrix extension method.

On an analogue of a property of singular M-matrices for the Lyapunov and Stein operators

A well-known result for a singular irreducible M-matrix A is that the only nonnegative vector that belongs to the range space of A is the zero vector. In this paper, we prove an analogue of this result for the Lyapunov and Stein transformations, which act on the inner product space of real symmetric matrices.

High-order exceptional points and novel light transmission spectra in [formula omitted] symmetric ring resonator array

In this paper we construct PT symmetric 1D multi-mode ring resonator arrays to study the high-order exceptional points and related light transmission spectra. We first focus on arrays with the underling symmetry given by SU(2) group possessing high-order exceptional points (EPs). The Gilmore-Perelomov coherent state approach has been well tested in SU(2) symmetric waveguide array, but how this approach is applied in resonator arrays, and how the divergent peaks change near high-order EPs is still remains to be seen. After the study on SU(2) symmetric arrays, we change the coupling coefficients to a uniformed value, and the high-order EP splits into several lower order EPs, thus the original clear boundary between the PT unbroken and broken regions is replaced by a transition region, wherein we observe some interesting results such as multi-number of non-divergent peaks and the coexistence of divergent and non-divergent peaks.

On spectral properties of digraphs about maximum distance

The maximum distance matrix of a strongly connected digraph is a symmetric matrix whose rows and columns are indexed the vertices, the entries of which correspond to the maximum directed distance between the vertices. In this paper, we determine the digraphs that uniquely minimize the largest eigenvalue of the maximum distance matrix in some classes of strongly connected digraphs, and the n-vertex strongly connected digraphs for which the maximum distance matrices have an eigenvalue with multiplicity n − 1.

A characterization of sets realizable by compensation in the SNIEP

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability.

Planar matrices and arrays of Feynman diagrams

Recently, planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k = 3 biadjoint amplitudes. Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition. In this work, we introduce planar matrices of Feynman diagrams as the objects that compute k = 4 biadjoint amplitudes. These are symmetric matrices of metric trees satisfying compatibility conditions. We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections. As applications of the first, we find all 693, 13 612 and 346 710 collections for (k, n) = (3, 7), (3, 8) and (3, 9), respectively. As applications of the second kind, we find all 90 608 and 30 659 424 planar matrices that compute (k, n) = (4, 8) and (4, 9) biadjoint amplitudes, respectively. As an example of the evaluation of matrices of Feynman diagrams, we present the complete form of the (4, 8) and (4, 9) biadjoint amplitudes. We also start a study of higher-dimensional arrays of Feynman diagrams, including the combinatorial version of the duality between (k, n) and (n − k, n) objects.

Changes in Community Structure of Brain Dynamic Functional Connectivity States in Mild Cognitive Impairment

Recent researches have noted many changes of short-term dynamic modalities in mild cognitive impairment (MCI) patients' brain functional networks. In this study, the dynamic functional brain networks of 82 MCI patients and 85 individuals in the normal control (NC) group were constructed using the sliding window method and Pearson correlation. The window size was determined using single-scale time-dependent (SSTD) method. Subsequently, k-means was applied to cluster all window samples, identifying three dynamic functional connectivity (DFC) states. Collective sparse symmetric non-negative matrix factorization (cssNMF) was then used to perform community detection on these states and quantify differences in brain regions. Finally, metrics such as within-community connectivity strength, community strength, and node diversity were calculated for further analysis. The results indicated high similarity between the two groups in state 2, with no significant differences in optimal community quantity and functional segregation (p < 0.05). However, for state 1 and state 3, the optimal community quantity was smaller in MCI patients compared to the NC group. In state 1, MCI patients had lower within-community connectivity strength and overall strength than the NC group, whereas state 3 showed results opposite to state 1. Brain regions with statistical difference included MFG.L, ORBinf.R, STG.R, IFGtriang.L, CUN.L, CUN.R, LING.R, SOG.L, and PCUN.R. This study on DFC states explores changes in the brain functional networks of patients with MCI from the perspective of alterations in the community structures of DFC states. The findings could provide new insights into the pathological changes in the brains of MCI patients.

A closer look at some new lower bounds on the minimum singular value of a matrix

There is an extensive body of literature on estimating the eigenvalues of the sum of two symmetric matrices, P+Q, in relation to the eigenvalues of P and Q. Recently, the authors introduced two novel lower bounds on the minimum eigenvalue, λmin(P+Q), under the conditions that matrices P and Q are symmetric positive semi-definite and their sum P+Q is non-singular. These bounds rely on the Friedrichs angle between the range spaces of matrices P and Q, which are denoted by R(P) and R(Q), respectively. In addition, both results led to the derivation of several new lower bounds on the minimum singular value of full-rank matrices. One significant aspect of the two novel lower bounds on λmin(P+Q) is the distinction of the case where R(P) and R(Q) have no principal angles between 0 and π2. This work offers an explanation for the aforementioned scenario and presents a classification of all matrices that meet the specified criteria. Additionally, we offer insight into the rationale behind selecting the decomposition for the subspace R(Q), which is employed to formulate the lower bounds for λmin(P+Q). At last, an example that showcases the potential for improving these two lower bounds is presented.

A Nyström-based low-rank unitary MVDR beamforming scheme

A Nyström-based scheme is devised to approximate the real symmetrical covariance matrix of the popular unitary MVDR (U-MVDR) beamformer with low-rank kernel matrix, based on which a novel lightweight and robust Nyström-based U-MVDR (NU-MVDR) beamformer is developed. Subsequently, the full-dimensional Nyström covariance matrix estimator’s signal subspace can be calculated by eigenvalue decomposition (EVD) of two reduced-dimensional matrices, which will save a significant amount of computing load at the EVD stage. This scheme has obvious computational efficiency and lightweight storage overheads. Meanwhile, we derive the effect of the Nyström covariance matrix’s rank on the output signal-to-interference-plus-noise ratio (SINR). Numerical results demonstrate that the proposed beamformer performs well in less snapshots and steering vector errors cases.

Secondary range symmetric matrices.

The concept of secondary range symmetric matrices is introduced here. Some characterizations as well as the equivalent conditions for a range symmetric matrix to be secondary range symmetric matrix is given. The idea of range symmetric matrices, range symmetric matrices over Minkowski space and secondary range symmetric matrices are different, and is depicted with the help of suitable examples. Finally, a necessary and sufficient condition for a secondary range symmetric matrix to have a secondary generalized inverse has been obtained.

On the stability of linear systems with a quadratic integral

The problem of stability of non-degenerate linear systems admitting a first integral in the form of a non-degenerate quadratic form is considered. New algebraic criteria for stability, as well as complete instability of such systems, have been established in the form of equality to zero of traces of products of matrices, which include an additional symmetric matrix. These conditions are closely related to the symplectic geometry of the phase space, which is determined by the matrix of the original linear system and the symmetric matrix defining the first integral. General results are applied to finding conditions for complete instability of linear gyroscopic systems.

Independence and orthogonality of algebraic eigenvectors over the max-plus algebra

The max-plus algebra R ∪ { − ∞ } is a semiring with the two operations: addition a ⊕ b := max ( a , b ) and multiplication a ⊗ b := a + b . Roots of the characteristic polynomial of a max-plus matrix are called algebraic eigenvalues. Recently, algebraic eigenvectors with respect to algebraic eigenvalues were introduced as a generalized concept of eigenvectors. In this paper, we present properties of algebraic eigenvectors analogous to those of eigenvectors in the conventional linear algebra. First, we prove that for generic matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are linearly independent. We further prove that algebraic eigenvectors of symmetric matrices with respect to distinct algebraic eigenvalues are orthogonal to each other.

Nodal Decompositions of a Symmetric Matrix

Abstract Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We show that for an arbitrary symmetric matrix, a positive fraction of eigenbases satisfy a generalized version of known nodal bounds for un-signed (that is classical) graphs. We do this through an explicit decomposition. Moreover, we show that with high probability, the number of nodal domains of a bulk eigenvector of the adjacency matrix of a signed Erdős-Rényi graph is $\Omega (n/\log n)$ and $o(n)$.

Task-oriented analysis and visualization of correlation patterns in multi-sensor time series

Time series data are usually collected through instruments equipped with sensors. For multi-sensor time series (MSTS), it is crucial to identify the factors that affect tasks such as classification and regression. To better understand the mechanisms affecting the task, we extracted the correlation patterns between sensors and represented them as symmetric positive definite matrices. The correlation patterns were then transformed into vector form to serve as input features for regression or classification models, and the models were trained for each sensor. Finally, we leveraged the interpretability of the models to analyze and visualize the correlation patterns at both micro and macro scales. By integrating the explanatory power of the models with correlation patterns, we could interpret the task in terms of time and space, providing valuable insights for exploring the underlying data rules. We evaluated our proposed method using both synthetic and real data, and the simulation results confirmed its effectiveness.

Three-stage transfer learning for motor imagery EEG recognition.

Motor imagery (MI) paradigms have been widely used in neural rehabilitation and drowsiness state assessment. The progress in brain-computer interface (BCI) technology has emphasized the importance of accurately and efficiently detecting motor imagery intentions from electroencephalogram (EEG). Despite the recent breakthroughs made in developing EEG-based algorithms for decoding MI, the accuracy and efficiency of these models remain limited by technical challenges posed by cross-subject heterogeneity in EEG data processing and the scarcity of EEG data for training. Inspired by the optimal transport theory, this study aims to develop a novel three-stage transfer learning (TSTL) method, which uses the existing labeled data from a source domain to improve classification performance on an unlabeled target domain. Notably, the proposed method comprises three components, namely, the Riemannian tangent space mapping (RTSM), source domain transformer (SDT), and optimal subspace mapping (OSM). The RTSM maps a symmetric positive definite matrix from the Riemannian space to the tangent space to minimize the marginal probability distribution drift. The SDT transforms the source domain to a target domain by finding the optimal transport mapping matrix to reduce the joint probability distribution differences. The OSM finally maps the transformed source domain and original target domain to the same subspace to further mitigate the distribution discrepancy. The performance of the proposed method was validated on two public BCI datasets, and the average accuracy of the algorithm on two datasets was 72.24% and 69.29%. Our results demonstrated the improved performance of EEG-based MI detection in comparison with state-of-the-art algorithms.

The decimation scheme for symmetric matrix factorization

Matrix factorization is an inference problem that has acquired importance due to its vast range of applications that go from dictionary learning to recommendation systems and machine learning with deep networks. The study of its fundamental statistical limits represents a true challenge, and despite a decade-long history of efforts in the community, there is still no closed formula able to describe its optimal performances in the case where the rank of the matrix scales linearly with its size. In the present paper, we study this extensive rank problem, extending the alternative ‘decimation’ procedure that was recently introduced by Camilli and Mézard, and carry out a thorough study of its performance. Decimation aims at recovering one column/line of the factors at a time, by mapping the problem into a sequence of neural network models of associative memory at a tunable temperature. The main advantage of decimation is that its theoretical performance can be studied using statistical physics techniques. In this paper we provide the general analysis applying to a large class of compactly supported priors and we show that the replica symmetric free entropy of the neural network models takes a universal form in the low temperature limit. We then study the decimation phase diagrams in two concrete cases, the sparse Ising prior and the uniform prior, for which we show that matrix factorization is theoretically possible when the ration P/N of rank to variable is below a certain threshold. This suggest that a possible route to solving algorithmically the matrix factorization problem could be to find efficient decimation algorithms; in this respect, we show that simple simulated annealing, though being effective for limited signal sizes, is not scalable.

A symmetric low-rank subspace clustering method for cooperative spectrum sensing in complex environments

A new cooperative spectrum sensing (CSS) method based on low-rank symmetric subspace clustering is proposed to improve spectrum sensing performance in complex environments. To align with the evolution of cellular networks and leverage spatial observations, a CSS model with multiple primary users and antennas is considered. To address the asymmetric weighting issue of data pairs, a low-rank representation with a symmetric constraint is proposed that reduces the signal dimensions. Finally, features are extracted from the symmetric coefficient matrix and clustered using the K-means algorithm. Through simulation, the effectiveness of the novel algorithm is numerically validated through comparative experiments.

Eigenphase distributions of unimodular circular ensembles

Abstract Motivated by the study of Polyakov lines in gauge theories, Hanada and Watanabe recently presented a conjectured formula for the distribution of eigenphases of Haar-distributed random SU(N) matrices (β = 2), supported by explicit examples at small N and by numerical samplings at larger N. In this letter, I spell out a concise proof of their formula, and present its orthogonal and symplectic counterparts, i.e. the eigenphase distributions of Haar-random unimodular symmetric (β = 1) and selfdual (β = 4) unitary matrices parametrizing SU(N)/SO(N) and SU(2N)/Sp(2N), respectively.

On the symmetry of photon detection arrays: A directionally sensitive 3D model

A detector's ability to obtain the direction of a radioactive source is an invaluable operational asset. A 2D/3D model was developed based on directionally sensitive arrays. The average location of photon interactions within a symmetrical array yields the direction of the source. The model is validated with simulations and laboratory measurements, maximum systematic error being 5–10° at energies >200 keV and approaching zero at lower energies. The symmetry model yields the direction of a shielded source even when no full energy photons could be detected.