7,622 results on '"Hermitian matrix"'
Search Results
2. Determinantal polynomials and the base polynomial of a square matrix over a finite field
- Author
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Edoardo Ballico
- Subjects
Finite field ,Hermitian matrix ,Base polynomial ,Numerical range ,Mathematics ,QA1-939 - Abstract
Purpose – The author studies forms over finite fields obtained as the determinant of Hermitian matrices and use these determinatal forms to define and study the base polynomial of a square matrix over a finite field. Design/methodology/approach – The authors give full proofs for the new results, quoting previous works by other authors in the proofs. In the introduction, the authors quoted related references. Findings – The authors get a few theorems, mainly describing some monic polynomial arising as a base polynomial of a square matrix. Originality/value – As far as the author knows, all the results are new, and the approach is also new.
- Published
- 2024
- Full Text
- View/download PDF
3. Linear codes associated to determinantal varieties in the space of hermitian matrices.
- Author
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Singh, Kanchan, Pathak, Ritesh Kumar, and Singh, Sheo Kumar
- Subjects
LINEAR codes ,FINITE fields - Abstract
We introduce a new class of linear codes over a finite field associated to determinantal varieties in the space of hermitian matrices and determine their length, dimension and minimum distance along with the weight spectrum. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. A New Extended Target Detection Method Based on the Maximum Eigenvalue of the Hermitian Matrix.
- Author
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Xu, Yong, Zhu, Yongfeng, and Song, Zhiyong
- Subjects
- *
MATRICES (Mathematics) , *EIGENVALUES , *RADAR cross sections , *CLUTTER (Radar) , *RADAR targets , *LIKELIHOOD ratio tests - Abstract
In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations.
- Author
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Zhang, Yue, Wang, Qing-Wen, and Xie, Lv-Ming
- Subjects
- *
QUATERNIONS , *EQUATIONS , *LEAST squares , *MATRICES (Mathematics) , *COMMUTATIVE algebra - Abstract
This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. A New Extended Target Detection Method Based on the Maximum Eigenvalue of the Hermitian Matrix
- Author
-
Yong Xu, Yongfeng Zhu, and Zhiyong Song
- Subjects
extended target ,range image ,Hermitian matrix ,eigenvalue of maximum ,detection ,Science - Abstract
In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets.
- Published
- 2024
- Full Text
- View/download PDF
7. Batched Eigenvalue Decomposition Algorithms for Hermitian Matrices on GPU
- Author
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HUANG Rongfeng, LIU Shifang, ZHAO Yonghua
- Subjects
hermitian matrix ,eigenvalue decomposition ,batch computing ,roofline model ,performance evaluation ,Computer software ,QA76.75-76.765 ,Technology (General) ,T1-995 - Abstract
Batched matrix computing problems are widely existed in scientific computing and engineering applications.With rapid performance improvements,GPU has become an important tool to solve such problems.The eigenvalue decomposition belongs to the two-sided decomposition and must be solved by the iterative algorithm.Iterative numbers for different matrices can be varied.Therefore,designing eigenvalue decomposition algorithms for batched matrices on the GPU is more challenging than designing batched algorithms for the one-sided decomposition,such as LU decomposition.This paper proposes batched algorithms based on the Jacobi algorithms for eigenvalue decomposition of Hermitian matrices.For matrices that cannot reside in shared memory wholly,the block technique is used to improve the arithmetic intensity,thus improving the use of GPU resources.Algorithms presented in this paper run completely on the GPU,avoiding the communication between the CPU and GPU.Kernel fusion is adopted to decrease the overhead of launching kernel and global memory access.Experimental results on V100 GPU show that our algorithms are better than existing works.Performance evaluation results of the Roofline model indicate that our implementations are close to the upper bound,approaching 4.11TFLOPS.
- Published
- 2023
- Full Text
- View/download PDF
8. Singular value and unitarily invariant norm inequalities for matrices
- Author
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Al-Natoor, Ahmad, Hirzallah, Omar, and Kittaneh, Fuad
- Published
- 2024
- Full Text
- View/download PDF
9. The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations
- Author
-
Yue Zhang, Qing-Wen Wang, and Lv-Ming Xie
- Subjects
commutative quaternion algebra ,matrix equations ,Hermitian matrix ,least squares solution ,Mathematics ,QA1-939 - Abstract
This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples.
- Published
- 2024
- Full Text
- View/download PDF
10. An estimation of HOMO–LUMO gap for a class of molecular graphs
- Author
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Hameed Saira, Alamer Ahmed, Javaid Muhammad, and Ahmad Uzma
- Subjects
molecular graph ,eigen spectrum ,homo–lumo gap ,bipartite graphs ,hermitian matrix ,Chemistry ,QD1-999 - Abstract
For any simple connected graph G of order n, having eigen spectrum μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n with middle eigenvalues μ H and μ L, where H = ⌊(n + 1)/2⌋ and L = ⌈(n + 1)/2⌉, the HOMO–LUMO gap is defined as as ΔG = μ H = μ L. In this article, a simple upper bound for the HOMO–LUMO gap corresponding to a special class of connected bipartite graphs is estimated. As an application, the upper bounds for the HOMO–LUMO gap of certain classes of nanotubes and nanotori are estimated.
- Published
- 2022
- Full Text
- View/download PDF
11. Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand.
- Author
-
Markovich, Liubov A., Migliore, Agostino, and Messina, Antonino
- Subjects
- *
ALGEBRAIC equations , *PARAMETERIZATION - Abstract
We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
12. Eigenvalues and diagonal elements.
- Author
-
Bhatia, Rajendra and Sharma, Rajesh
- Abstract
A basic theorem in linear algebra says that if the eigenvalues and the diagonal entries of a Hermitian matrix A are ordered as λ 1 ≤ λ 2 ≤... ≤ λ n and a 1 ≤ a 2 ≤... ≤ a n , respectively, then λ 1 ≤ a 1 . We show that for some special classes of Hermitian matrices this can be extended to inequalities of the form λ k ≤ a 2 k - 1 , k = 1 , 2 ,... , ⌈ n 2 ⌉ . [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
13. On the Cospectrality of Hermitian Adjacency Matrices of Mixed Graphs.
- Author
-
Alomari, Omar, Abudayah, Mohammad, and Ghanem, Manal
- Subjects
- *
MATRICES (Mathematics) , *CHARTS, diagrams, etc. - Abstract
A mixed graph D is a graph that can be obtained from a graph by orienting some of its edges. Let α be a primitive nth root of unity, then the α−Hermitian adjacency matrix of a mixed graph is defined to be the matrix Hα = [hrs] where hrs = α if rs is an arc in D, hrs = α if sr is an arc in D, hrs = 1 if sr is a digon in D and hrs = 0 otherwise. In this paper we study the cospectrality of the Hermitian adjacency matrix of a mixed graph. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
14. Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand
- Author
-
Liubov A. Markovich, Agostino Migliore, and Antonino Messina
- Subjects
companion matrix ,almost-companion matrix ,hermitian matrix ,unitary matrix ,complex polynomial ,density matrix ,Science ,Astrophysics ,QB460-466 ,Physics ,QC1-999 - Abstract
We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees.
- Published
- 2023
- Full Text
- View/download PDF
15. Some inequalities involving eigenvalues and positive linear maps
- Author
-
Kumar, Ravinder and Sharma, Rajesh
- Published
- 2023
- Full Text
- View/download PDF
16. High-Throughput FPGA Implementation of Matrix Inversion for Control Systems.
- Author
-
Zhang, Xiao-Wei, Zuo, Lei, Li, Ming, and Guo, Jian-Xin
- Subjects
- *
PHASED array radar , *MATRIX decomposition , *MATRIX multiplications , *FIELD programmable gate arrays , *GATE array circuits , *MATRIX inversion - Abstract
In control engineering, numerical stability and real time are the two most important requirements for the matrix inversion. This article presents an efficient and robust method for the field-programmable gate array (FPGA) calculation of the matrix inversion. We initially consider the scenario that the matrix to be processed is a nonsingular Hermitian matrix. The proposed computation procedures are composed of the matrix decomposition, triangular matrix inversion, and matrices multiplication. The first procedure is completed by LDL factorization based on the outer form of Cholesky's method, whereas the recursive algorithm for block submatrices is adopted to achieve the triangular matrix inversion. The new method has the high level in the parallel pipelining mechanism and steals the characteristics of both the upper triangular matrix and its inversion to reduce the computation load and improve the numerical stability. Furthermore, the non-Hermitian matrix inversion can be easily solved if another procedure is added in the new method. Finally, we compare our method with the exiting FPGA-based techniques on one Xilinx Virtex-7 XC7VX690T FPGA. Meanwhile, it has solved one array antenna control problem of the adaptive digital beam forming for one phased array radar successfully. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
17. New Hilbert–Schmidt norm inequalities for positive semidefinite matrices
- Author
-
Hayajneh, Mostafa, Hayajneh, Saja, and Kittaneh, Fuad
- Published
- 2023
- Full Text
- View/download PDF
18. Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand
- Abstract
We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees., QID/Borregaard Group
- Published
- 2023
- Full Text
- View/download PDF
19. A Relation Between Moore-Penrose Inverses of Hermitian Matrices and Its Application in Electrical Networks
- Author
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Yujun Yang, Dayong Wang, and Douglas J. Klein
- Subjects
resistance distance ,electrical network ,Hermitian matrix ,Laplacian matrix ,Moore-Penrose inverse ,Physics ,QC1-999 - Abstract
A novel relation between the Moore-Penrose inverses of two nullity-1 n × n Hermitian matrices which share a common null eigenvector is established, and its application in electrical networks is illustrated by applying the result to Laplacian matrices of graphs.
- Published
- 2020
- Full Text
- View/download PDF
20. Some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices
- Author
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Yan Hong, Dongkyu Lim, and Feng Qi
- Subjects
Generalized eigenvalue ,Hermitian matrix ,Inequality ,Perturbation problem ,Mathematics ,QA1-939 - Abstract
Abstract In the paper, the authors establish some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices and modify shortcomings of some known inequalities for generalized eigenvalues in the related literature.
- Published
- 2018
- Full Text
- View/download PDF
21. On some open questions concerning determinantal inequalities.
- Author
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Ghabries, Mohammad M., Abbas, Hassane, and Mourad, Bassam
- Subjects
- *
OPEN-ended questions , *MATHEMATICAL equivalence - Abstract
In 2017, M. Lin formulated two conjectures concerning determinantal inequalities for positive semi-definite matrices A and B , and which can be stated as follows det (A 2 + | A B | p) ≥ det (A 2 + | B A | p) for p ≥ 0 and det (A 2 + | A B | p) ≥ det (A 2 + A p B p) for 0 ≤ p ≤ 2. The main goal of this paper is to confirm the first conjecture in a slightly more general setting namely in the case when A and B are Hermitian, and also to prove the second conjecture when 0 ≤ p ≤ 4 3. Various related inequalities are then presented and we conclude with an open log-majorization question. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
22. Some lower bounds for the energy of graphs.
- Author
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Akbari, Saieed, Ghodrati, Amir Hossein, and Hosseinzadeh, Mohammad Ali
- Subjects
- *
GRAPH connectivity , *BIPARTITE graphs , *ABSOLUTE value , *EIGENVALUES , *SQUARE root - Abstract
The singular values of a matrix A are defined as the square roots of the eigenvalues of A ⁎ A , and the energy of A denoted by E (A) is the sum of its singular values. The energy of a graph G , E (G) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = ( B D D ⁎ C ) , then E (A) ≥ 2 E (D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E (H) is an edge cut of G , then E (H) ≤ E (G) , i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E (G) ≥ 4 m + n (n − 2) | det (A) | 2 n . Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E (G) given in Oboudi (2019) [14]. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
23. Bordered Hermitian matrices and sums of the Möbius function.
- Author
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Kline, Jeffery
- Subjects
- *
MOBIUS function , *PRIME number theorem , *MATRICES (Mathematics) , *MATHEMATICAL convolutions - Abstract
Matrices in R n × n with determinant equal to ∑ i ≤ n μ (i) , where μ represents the Möbius function, have been studied for decades. The motivation to study such matrices is the close connection that they have to the prime number theorem, Dirichlet convolution, and related concepts. We introduce two parameterized families of bordered Hermitian matrices that possess similar properties. Each family is comprised of matrices M s ∈ C (n − 1) × (n − 1) that satisfy det M 0 = ∑ i ≤ n | μ (i) | , det M 1 = (∑ i ≤ n μ (i)) 2 , and we show det M s is a quadratic polynomial in s. We apply the Cauchy interlacing theorem to show that, for each matrix in one of the families, the product of all of the subdominant eigenvalues is bounded above by 6 / π 2 + O (n − 1 / 2). [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
24. NEW BOUNDS FOR THE SPREAD OF A MATRIX USING THE RADIUS OF THE SMALLEST DISC THAT CONTAINS ALL EIGENVALUES.
- Author
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FRAKIS, A.
- Subjects
- *
RADIUS (Geometry) , *MATRICES (Mathematics) , *EIGENVALUES , *EVIDENCE , *MATHEMATICAL equivalence - Abstract
Let D denote the smallest disc containing all eigenvalues of the matrix A. Without knowing the eigenvalues of A, we can estimate the spread of A and the radius of D. Some new bounds for the radius of D and the spread of A are given. These bounds involve the entries of A. Also sufficient conditions for equality are obtained for some inequalities. New proofs of some known results are presented, too. [ABSTRACT FROM AUTHOR]
- Published
- 2020
25. The Jordan Canonical Form of a Product of Elementary S-unitary Matrices.
- Author
-
Gonda, Erwin J. and Paras, Agnes T.
- Subjects
- *
COMPLEX matrices , *MATRICES (Mathematics) - Abstract
Let S be an n-by-n, nonsingular, and Hermitian matrix. A square complex matrix Q is said to be S-unitary if Q*SQ = S. An S-unitary matrix Q is said to be elementary if rank(Q -- I) = 1. It is known what form every elementary S-unitary can take, and that every S-unitary can be written as a product of elementary S-unitaries. In this paper, we determine the Jordan canonical form of a product of two elementary S-unitaries. [ABSTRACT FROM AUTHOR]
- Published
- 2020
26. Making almost commuting matrices commute
- Author
-
Hastings, Matthew [Los Alamos National Laboratory]
- Published
- 2008
27. More about chiral symmetry restoration at finite temperature in the planar limit
- Author
-
Neuberger, H
- Published
- 2007
- Full Text
- View/download PDF
28. Finite boson mappings of fermion systems
- Author
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Ginocchio, J
- Published
- 1994
29. From the shell model to the interacting boson model
- Author
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Johnson, C
- Published
- 1994
30. Weyl expansion for multicomponent wave equations
- Author
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Andre, Daniel Batista
- Subjects
530.1 ,QUANTUM MECHANICS ,WEYL UNIFIED THEORY ,HAMILTONIANS ,HERMITIAN MATRIX ,MATHEMATICAL MODELS - Published
- 2000
31. Hermitian and unitary almost-companion matrices of polynomials on demand
- Author
-
Antonino MESSINA, Agostino Migliore, and Liubov Markovich
- Subjects
almost-companion matrix ,Quantum Physics ,companion matrix ,hermitian matrix ,unitary matrix ,complex polynomial ,density matrix ,sub-parameterization ,General Physics and Astronomy ,FOS: Physical sciences ,Quantum Physics (quant-ph) - Abstract
We introduce the concept of Almost-Companion Matrix (ACM) by relaxing the non-derogatory property of the standard Companion Matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and Unitary ACMs starting from appropriate third degree polynomials, with implications for their use in physical-mathematical problems such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degree., Comment: 26 pages
- Published
- 2023
- Full Text
- View/download PDF
32. The converse of Weyl's eigenvalue inequality.
- Author
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Wang, Yi and Zheng, Sainan
- Subjects
- *
MATHEMATICAL equivalence , *MATRICES (Mathematics) - Abstract
We establish the converse of Weyl's eigenvalue inequality for additive Hermitian perturbations of a Hermitian matrix. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
33. The change in multiplicity of an eigenvalue due to adding or removing edges.
- Author
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Johnson, Charles R., Saiago, Carlos M., and Toyonaga, Kenji
- Subjects
- *
EIGENVALUES , *EDGES (Geometry) , *HERMITIAN forms , *GEOMETRIC vertices , *GRAPHIC methods - Abstract
Abstract We investigate the change in the multiplicities of the eigenvalues of a Hermitian matrix with a simple graph G , when edges are inserted into G or removed from G. We focus upon cases in which the multiplicity of the eigenvalue does not change due to inserting or removing edges incident to a vertex. Furthermore, we show how the change in the multiplicities of the eigenvalues occur, when two disjoint graphs are connected with one edge, based upon the status of the vertices that are connected. Lastly, we give the possible classifications of cut-edges in a graph and characterize the occurrence of each possible status. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
34. Hill representations for ∗-linear matrix maps
- Author
-
A. van der Merwe and S. ter Horst
- Subjects
Combinatorics ,Linear map ,Matrix (mathematics) ,General Mathematics ,Nonnegative matrix ,Linear matrix ,Hermitian matrix ,Mathematics - Abstract
In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L : ℂ q × q → ℂ n × n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L ( V ∗ ) = L ( V ) ∗ , via representations of the form L ( V ) = ∑ k , l = 1 m H k l A l V A k ∗ , V ∈ ℂ q × q , for matrices A 1 , … , A m ∈ ℂ n × q and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A 1 , … , A m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H = H k l explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗ -linearity on the linear map L .
- Published
- 2022
35. Erasures Repair for Decreasing Monomial-Cartesian and Augmented Reed-Muller Codes of High Rate
- Author
-
Daniel Valvo, Hiram H. López, and Gretchen L. Matthews
- Subjects
FOS: Computer and information sciences ,Discrete mathematics ,High rate ,Monomial ,Computer science ,Information Theory (cs.IT) ,Computer Science - Information Theory ,Bandwidth (signal processing) ,Reed–Muller code ,020206 networking & telecommunications ,Data_CODINGANDINFORMATIONTHEORY ,02 engineering and technology ,Library and Information Sciences ,Hermitian matrix ,Computer Science Applications ,law.invention ,Dimension (vector space) ,law ,Scheme (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,Cartesian coordinate system ,11T71, 14G50 ,Computer Science::Information Theory ,Information Systems - Abstract
In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes DM-CC, a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.
- Published
- 2022
36. The Eigenvector-Eigenvalue Identity Applied to Fast Calculation of polSAR Scattering Characterization
- Author
-
Allan Nielsen
- Subjects
Complex covariance matrix ,Coherency matrix ,Entropy ,Mean alpha angle (α¯) ,Anisotropy ,Polarimetric SAR ,F-SAR ,X-band ,Hermitian matrix ,Electrical and Electronic Engineering ,Geotechnical Engineering and Engineering Geology - Abstract
Unlike the original Cloude-van Zyl decomposition of reflection symmetric polSAR data, a recently suggested version of the decomposition for full/quad pol data relies on the Cloude-Pottier mean alpha angle (ᾱ) to characterize the scattering mechanism. ᾱ can be calculated from the eigenvectors of the coherency matrix. By means of the eigenvector-eigenvalue identity (EEI) we can avoid the calculation of the eigenvectors. The EEI finds ᾱ by means of eigenvalues of the 3×3 coherency matrix and its 2×2 minor(s) only and is well suited for fast array based computer implementation. In this paper with focus on computational aspects we demonstrate fast EEI based determination of ᾱ on X-band F-SAR image data over Vejers, Denmark, including a detailed example of calculations and computer code.
- Published
- 2022
37. Dirac series for E6(−14)
- Author
-
Lin-Gen Ding, Chao-Ping Dong, and Haian He
- Subjects
Pure mathematics ,Algebra and Number Theory ,Series (mathematics) ,Infinitesimal ,Simple Lie group ,Dirac (software) ,Type (model theory) ,Hermitian matrix ,Unitary state ,Cohomology ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics - Representation Theory ,Mathematics - Abstract
Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the simple Lie group $E_{6(-14)}$, which is of Hermitian symmetric type. Each FS-scattered Dirac series of $E_{6(-14)}$ is realized as a composition factor of certain $A_{\mathfrak{q}}(\lambda)$ module. Along the way, we have also obtained all the fully supported irreducible unitary representations of $E_{6(-14)}$ with integral infinitesimal characters., Comment: 32 pages, some strings are folded
- Published
- 2022
38. Determination of eigenvectors with Lagrange multipliers
- Author
-
Han, Wooyong, Jung, Dong-Won, Lee, Jungil, and Yu, Chaehyun
- Published
- 2021
- Full Text
- View/download PDF
39. Central Limit Theorem for Linear Eigenvalue Statistics of <scp>Non‐Hermitian</scp> Random Matrices
- Author
-
Dominik Schröder, László Erdős, and Giorgio Cipolloni
- Subjects
Independent and identically distributed random variables ,Applied Mathematics ,General Mathematics ,Gaussian ,Hermitian matrix ,symbols.namesake ,Matrix (mathematics) ,Distribution (mathematics) ,Statistics ,symbols ,Random matrix ,Eigenvalues and eigenvectors ,Central limit theorem ,Mathematics - Abstract
We consider large non-Hermitian random matrices $X$ with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having $2+\epsilon$ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [Rider, Silverstein 2006], or the distribution of the matrix elements needed to be Gaussian [Rider, Virag 2007], or at least match the Gaussian up to the first four moments [Tao, Vu 2016; Kopel 2015]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of $X$ with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian Motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices $X$ that are presented in the companion paper [Cipolloni, Erdős, Schroder 2019].
- Published
- 2021
40. Orbit spaces for torus actions on Hessenberg varieties
- Author
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Vladislav Vladimirovich Cherepanov
- Subjects
Pure mathematics ,Algebra and Number Theory ,Isospectral ,Torus ,Orbit (control theory) ,Fixed point ,Space (mathematics) ,Hermitian matrix ,Manifold ,Hessenberg variety ,Mathematics - Abstract
In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to the union of disjoint open subsets of the -sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.
- Published
- 2021
41. Impossibility Results for Constrained Control of Stochastic Systems
- Author
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Masako Kishida and Ahmet Cetinkaya
- Subjects
Moment (mathematics) ,Noise ,Control and Systems Engineering ,Computer science ,Control theory ,Bounded function ,Linear system ,Second moment of area ,Networked control system ,State (functional analysis) ,Electrical and Electronic Engineering ,Hermitian matrix ,Computer Science Applications - Abstract
Strictly unstable linear systems under additive and nonvanishing stochastic noise with unbounded supports are known to be impossible to stabilize by using deterministically constrained control inputs. In this paper, similar impossibility results are obtained for the scenarios where the control input is probabilistically constrained and the support of the noise distribution is not necessarily unbounded. In particular, control policies that have bounded time-averaged second moments are considered. It is shown that for such control policies, there are critical average moment bounds, below which second moment stabilization of a linear stochastic system is not possible, and moreover, second moment of the state diverges regardless of the choice of control policy and the initial state distribution. Nonnegative-definite Hermitian matrices are exploited to extract sufficient instability conditions that can be assessed by using the eigenstructure of the system matrix and the distribution of the noise. The results indicate that in certain networked control system settings with noise, designing stabilizing constrained controllers is an impossible task, if the probability of successful transmissions of control commands over the network is known to be too small in average.
- Published
- 2021
42. Rank and Kernel of Additive Generalized Hadamard Codes
- Author
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Steven T. Dougherty, Josep Rifà, and Mercè Villanueva
- Subjects
Trace (linear algebra) ,Kernel (set theory) ,Rank (linear algebra) ,Linear space ,Dimension (graph theory) ,Generalised Hadamard matrix ,Library and Information Sciences ,Rank ,Upper and lower bounds ,Hermitian matrix ,Computer Science Applications ,Combinatorics ,Kernel ,Generalised Hadamard code ,Product (mathematics) ,Additive code ,Nonlinear code ,Information Systems ,Mathematics - Abstract
L'article pertany al grup de recerca Combinatorics, Coding and Security Group (CCSG) A subset of a vector space Fn q is additive if it is a linear space over the field Fp, where q = pe, p prime, and e > 1. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case e = 2, it is shown that the given bounds are tight and it is possible to construct an additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.
- Published
- 2021
43. Non-Hermitian physics for optical manipulation uncovers inherent instability of large clusters
- Author
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Xiao Li, Jack C. Ng, Yineng Liu, Zhifang Lin, and Che Ting Chan
- Subjects
Physics ,Physics::General Physics ,Multidisciplinary ,Operator (physics) ,Science ,Time evolution ,General Physics and Astronomy ,General Chemistry ,Instability ,Hermitian matrix ,General Biochemistry, Genetics and Molecular Biology ,Article ,symbols.namesake ,Classical mechanics ,Optical physics ,Optical tweezers ,Optical manipulation and tweezers ,Dissipative system ,symbols ,Cluster (physics) ,Lorentz force - Abstract
Intense light traps and binds small particles, offering unique control to the microscopic world. With incoming illumination and radiative losses, optical forces are inherently nonconservative, thus non-Hermitian. Contrary to conventional systems, the operator governing time evolution is real and asymmetric (i.e., non-Hermitian), which inevitably yield complex eigenvalues when driven beyond the exceptional points, where light pumps in energy that eventually “melts” the light-bound structures. Surprisingly, unstable complex eigenvalues are prevalent for clusters with ~10 or more particles, and in the many-particle limit, their presence is inevitable. As such, optical forces alone fail to bind a large cluster. Our conclusion does not contradict with the observation of large optically-bound cluster in a fluid, where the ambient damping can take away the excess energy and restore the stability. The non-Hermitian theory overturns the understanding of optical trapping and binding, and unveils the critical role played by non-Hermiticity and exceptional points, paving the way for large-scale manipulation., Non-conservativeness plays a mysterious role in optical trapping. Applying the non-Hermitian theory, the authors showed that the existence of exceptional points drives the Lorentz force to lose its ability to bind clusters of ~10 or more microparticles, unless remedied by dissipative forces.
- Published
- 2021
44. Galois self-orthogonal constacyclic codes over finite fields
- Author
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Hongwei Liu and Yuqing Fu
- Subjects
Combinatorics ,Finite field ,Integer ,Applied Mathematics ,Product (mathematics) ,Euclidean geometry ,Hermitian matrix ,Prime (order theory) ,Computer Science Applications ,Vector space ,Mathematics - Abstract
Let $${\mathbb {F}}_{q}$$ be a finite field with $$q=p^{e}$$ elements, where p is a prime and e is a positive integer. In 2017, Fan and Zhang introduced $$\ell $$ -Galois inner products on the n-dimensional vector space $${\mathbb {F}}_{q}^{n}$$ for $$0\le \ell
- Published
- 2021
45. Integral binary Hamiltonian forms and their waterworlds
- Author
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Jouni Parkkonen, Frédéric Paulin, University of Jyväskylä (JYU), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and PICS 6950 CNRS
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Binary number ,01 natural sciences ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,waterworld ,differentiaaligeometria ,maximal order ,hyperbolic 5-space ,0103 physical sciences ,0101 mathematics ,Algebraic number ,reduction theory ,Mathematics ,lukuteoria ,Mathematics - Number Theory ,Quaternion algebra ,010102 general mathematics ,Hamilton-Bianchi group ,ryhmäteoria ,Order (ring theory) ,Mathematics::Geometric Topology ,Hermitian matrix ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Binary quadratic form ,010307 mathematical physics ,Geometry and Topology ,rational quaternion algebra ,Mathematics - Group Theory ,binary Hamiltonian form ,Hamiltonian (control theory) - Abstract
We give a graphical theory of integral indefinite binary Hamiltonian forms $f$ analogous to the one by Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathcal O$ in a definite quaternion algebra over $\mathbb Q$, we define the waterworld of $f$, analogous to Conway's river and Bestvina-Savin's ocean, and use it to give a combinatorial description of the values of $f$ on $\mathcal O\times\mathcal O$. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), and the $\operatorname{SL}_2(\mathcal O)$-equivariant Ford-Voronoi cellulation of the real hyperbolic $5$-space., Comment: Revised version, 40 pages
- Published
- 2021
46. Topological complex-energy braiding of non-Hermitian bands
- Author
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Avik Dutt, Shanhui Fan, Kai Wang, and Charles C. Wojcik
- Subjects
Physics ,Ring (mathematics) ,Multidisciplinary ,Hopf link ,Braid group ,Braid ,Topology ,Unknot ,Mathematics::Geometric Topology ,Hermitian matrix ,Topology (chemistry) ,Unlink - Abstract
Effects connected with the mathematical theory of knots1 emerge in many areas of science, from physics2,3 to biology4. Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems5, where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹2. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension6,7. Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems. Experiments using two coupled optical ring resonators and based on the concept of synthetic dimension reveal non-Hermitian energy band structures exhibiting topologically non-trivial knots and links.
- Published
- 2021
47. On Hulls of Some Primitive BCH Codes and Self-Orthogonal Codes
- Author
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Chunyu Gan, Sihem Mesnager, Chengju Li, and Haifeng Qian
- Subjects
Dimension (graph theory) ,Order (ring theory) ,Library and Information Sciences ,Type (model theory) ,Hermitian matrix ,Upper and lower bounds ,Linear code ,Computer Science Applications ,Combinatorics ,Finite field ,Mathematics::Metric Geometry ,BCH code ,Information Systems ,Mathematics - Abstract
Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let $\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ be the primitive BCH code over $\mathbb {F}_{r}$ of length $r^{m}-1$ with designed distance $\delta $ , where $\mathbb {F}_{r}$ is the finite field of order $r$ . In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code $\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ and its hull in both Hermitian and Euclidean cases for $2 \le \delta \le r^{2}-1$ . We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.
- Published
- 2021
48. CLASSIFICATION OF 3-GRADED CAUSAL SUBALGEBRAS OF REAL SIMPLE LIE ALGEBRAS
- Author
-
Daniel Oeh
- Subjects
Algebra and Number Theory ,Endomorphism ,Direct sum ,010102 general mathematics ,Mathematics - Operator Algebras ,010103 numerical & computational mathematics ,Type (model theory) ,Automorphism ,01 natural sciences ,Hermitian matrix ,Linear subspace ,Combinatorics ,Lie algebra ,FOS: Mathematics ,Primary 22E45, Secondary 81R05, 81T05 ,Geometry and Topology ,0101 mathematics ,Invariant (mathematics) ,Representation Theory (math.RT) ,ddc:510 ,Mathematics::Representation Theory ,Operator Algebras (math.OA) ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $(\mathfrak{g},\tau)$ be a real simple symmetric Lie algebra and let $W \subset \mathfrak{g}$ be an invariant closed convex cone which is pointed and generating with $\tau(W) = -W$. For elements $h \in \mathfrak{g}$ with $\tau(h) = h$, we classify the Lie algebras $\mathfrak{g}(W,\tau,h)$ which are generated by the closed convex cones \[C_{\pm}(W,\tau,h) := (\pm W) \cap \mathfrak{g}_{\pm 1}^{-\tau}(h),\] where $\mathfrak{g}^{-\tau}_{\pm 1}(h) := \{x \in \mathfrak{g} : \tau(x) = -x, [h,x] = \pm x\}$. These cones occur naturally as the skew-symmetric parts of the Lie wedges of endomorphism semigroups of certain standard subspaces. We prove in particular that, if $\mathfrak{g}(W,\tau,h)$ is non-trivial, then it is either a hermitian simple Lie algebra of tube type or a direct sum of two Lie algebras of this type. Moreover, we give for each hermitian simple Lie algebra and each equivalence class of involutive automorphisms $\tau$ of $\mathfrak{g}$ with $\tau(W) = -W$ a list of possible subalgebras $\mathfrak{g}(W,\tau,h)$ up to isomorphy., Comment: The title of the paper has been changed; the introduction has been rewritten; some of the proofs have been shortened
- Published
- 2022
49. On stability of the fibres of Hopf surfaces as harmonic maps and minimal surfaces
- Author
-
Liding Huang and Jingyi Chen
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Minimal surface ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Hopf surface ,Harmonic map ,Harmonic (mathematics) ,Torus ,Hermitian matrix ,Cohomology ,Differential Geometry (math.DG) ,FOS: Mathematics ,Mathematics - Abstract
We construct a family of Hermitian metrics on the Hopf surface $ \mathbb{S}^3\times \mathbb{S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally K\"ahler. Among the toric fibres of $\pi:\mathbb{S}^{3} \times \mathbb{S}^1\to\mathbb{C} P^1$ two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces., Comment: We generalize Theorem 1.2 to $\mathbb{S}^{2n-1}\times\mathbb{S}^1$ in section 4
- Published
- 2021
50. Kudla–Rapoport cycles and derivatives of local densities
- Author
-
Wei Zhang and Chao Li
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Dimension (graph theory) ,Hermitian matrix ,Unitary state ,Identity (mathematics) ,symbols.namesake ,Mathematics::Algebraic Geometry ,Intersection ,Eisenstein series ,symbols ,Mathematics::Representation Theory ,Fourier series ,Mathematics - Abstract
We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia–Sankaran, we also prove cases of the arithmetic Siegel–Weil formula in any dimension.
- Published
- 2021
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