2,091 results on '"Unitary matrix"'
Search Results
2. A Unitary Transformation Extension of PolSAR Four-Component Target Decomposition.
- Author
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Wang, Tingting, Suo, Zhiyong, Ti, Jingjing, Yan, Boya, Xiang, Hongli, and Xi, Jiabao
- Subjects
- *
UNITARY transformations , *JACOBI method , *T-matrix , *DECOMPOSITION method , *MATRIX decomposition - Abstract
As an improvement on the traditional model-based Yamaguchi four-component decomposition method, in recent years, to fully utilize the polarization information in the coherency matrix, four-component target decomposition methods Y4R and S4R have been proposed, which are based on the rotation of the coherency matrix and the expansion of the volume model, respectively. At the same time, there is also an improved G4U method proposed based on Y4R and S4R. Although these methods have achieved certain decomposition results, there are still problems with overestimation of volume scattering and insufficient utilization of polarization information. In this paper, a unitary transformation extension to the four-component target decomposition method of PolSAR based on the properties of the Jacobi method is proposed. By analyzing the terms in the basic scattering models, such as volume scattering, in the existing four-component decomposition methods, it is clear that the reason for the existence of the residual matrix in the existing decomposition methods is that the off-diagonal term T 13 and the real part of T 23 of the coherency matrix T do not participate in the four-component decomposition. On this basis, a matrix transformation method is proposed to decouple terms T 13 and Re T 23 , and the residual matrix decomposed based on this method is derived. The performance of the proposed method was validated and evaluated using two datasets. The experimental results indicate that, compared with model-based methods such as Y4R, S4R and G4U, the proposed method can enhance the contribution of double-bounce scattering and odd-bounce scattering power in urban areas in both sets of data. The computational time of the proposed method is equivalent to Y4R, S4R, etc. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Design of a photonic unitary neural network based on MZI arrays.
- Author
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YE ZHANG, RUITING WANG, YEJIN ZHANG, YANMEI SU, PENGFEI WANG, GUANGZHEN LUO, XULIANG ZHOU, and JIAOQING PAN
- Subjects
- *
OPTICAL interferometers , *PHOTONICS , *BANDWIDTHS , *CLASSIFICATION , *ATTENTION - Abstract
In recent years, optical neural networks have attracted widespread attention, due to their advantages of high speed, high parallelism, high bandwidth, and low power consumption. Photonic unitary neural network is a kind of neural networks that utilize the principles of unitary matrices and photonics to perform computations. In this paper, we design a photonic unitary neural network based on Mach-Zehnder interferometer arrays. The results show that the network has a good performance on both triangular and circular binary classification datasets, where most of the data points are correctly classified. The accuracies achieve 97% and 95% for triangular and circular datasets, with the loss function values of 0.023 and 0.046, respectively. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. Unitary inner product graphs and their automorphisms.
- Author
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Zhao, Shouxiang and Nan, Jizhu
- Subjects
- *
FINITE fields , *ORBITS (Astronomy) , *AUTOMORPHISM groups , *AUTOMORPHISMS , *MORPHISMS (Mathematics) - Abstract
Let q 2 be a finite field of order q 2 and 2 ν + δ ≥ 2 be an integer with δ = 0 or 1 , where q is a power of a prime. We introduce the concept of the unitary inner product graph U i (2 ν + δ , q 2) over q 2 and determine its automorphism group. We obtain two necessary and sufficient conditions for two vertices of U i (2 ν + δ , q 2) and two edges of U i (2 ν + δ , q 2) , respectively, are in the same orbit under the action of the automorphism group of U i (2 ν + δ , q 2). [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
5. 四元数Lyapunov方程的酉结构解及最佳逼近.
- Author
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黄敬频 and 刘广梅
- Subjects
MATRIX decomposition ,QUATERNIONS ,EIGENVALUES ,MATRIX inequalities ,EQUATIONS - Abstract
Copyright of Journal of Chongqing University of Technology (Natural Science) is the property of Chongqing University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2023
- Full Text
- View/download PDF
6. A Unitary Transformation Extension of PolSAR Four-Component Target Decomposition
- Author
-
Tingting Wang, Zhiyong Suo, Jingjing Ti, Boya Yan, Hongli Xiang, and Jiabao Xi
- Subjects
model-based decomposition ,polarimetric synthetic aperture radar (PolSAR) ,unitary matrix ,target decomposition ,Jacobi method ,Science - Abstract
As an improvement on the traditional model-based Yamaguchi four-component decomposition method, in recent years, to fully utilize the polarization information in the coherency matrix, four-component target decomposition methods Y4R and S4R have been proposed, which are based on the rotation of the coherency matrix and the expansion of the volume model, respectively. At the same time, there is also an improved G4U method proposed based on Y4R and S4R. Although these methods have achieved certain decomposition results, there are still problems with overestimation of volume scattering and insufficient utilization of polarization information. In this paper, a unitary transformation extension to the four-component target decomposition method of PolSAR based on the properties of the Jacobi method is proposed. By analyzing the terms in the basic scattering models, such as volume scattering, in the existing four-component decomposition methods, it is clear that the reason for the existence of the residual matrix in the existing decomposition methods is that the off-diagonal term T13 and the real part of T23 of the coherency matrix T do not participate in the four-component decomposition. On this basis, a matrix transformation method is proposed to decouple terms T13 and ReT23, and the residual matrix decomposed based on this method is derived. The performance of the proposed method was validated and evaluated using two datasets. The experimental results indicate that, compared with model-based methods such as Y4R, S4R and G4U, the proposed method can enhance the contribution of double-bounce scattering and odd-bounce scattering power in urban areas in both sets of data. The computational time of the proposed method is equivalent to Y4R, S4R, etc.
- Published
- 2024
- Full Text
- View/download PDF
7. Reversible Computation in Integrated Photonics
- Author
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De Vos, Alexis, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Mezzina, Claudio Antares, editor, and Podlaski, Krzysztof, editor
- Published
- 2022
- Full Text
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8. Eigenvalue, Matrix Diagonalization and Unitary Matrix
- Author
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Wong, Hiu Yung and Wong, Hiu Yung
- Published
- 2022
- Full Text
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9. Real logarithms of semi-simple matrices
- Author
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Pertici, Donato
- Published
- 2023
- Full Text
- View/download PDF
10. On Normal and Binormal Matrices.
- Author
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Ikramov, Kh. D.
- Abstract
The problem discussed is how to obtain a normal matrix from a binormal one and, conversely, a binormal matrix from a normal one via the right multiplication on a suitable unitary matrix. Let be a normal matrix badly conditioned with respect to inversion, that is, having a large condition number . We show that, among the binormal matrices that can be obtained from , there is a matrix whose eigenvalues have individual condition numbers of order . [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
11. Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand.
- Author
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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. Conjectured DXZ decompositions of a unitary matrix.
- Author
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De Vos, Alexis, Idel, Martin, and De Baerdemacker, Stijn
- Subjects
- *
MATRIX decomposition - Abstract
For any unitary matrix there exists a ZXZ decomposition, according to a theorem by Idel and Wolf. For any even-dimensional unitary matrix there exists a block-ZXZ decomposition, according to a theorem by Führ and Rzeszotnik. We conjecture that these two decompositions are merely special cases of a set of decompositions, one for every divisor of the matrix dimension. For lack of a proof, we provide an iterative Sinkhorn algorithm to find an approximate numerical decomposition. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
13. On Genetic Unitary Matrices and Quantum-Algorithmic Genetics
- Author
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Petoukhov, Sergey V., Petukhova, Elena S., Kacprzyk, Janusz, Series Editor, Pal, Nikhil R., Advisory Editor, Bello Perez, Rafael, Advisory Editor, Corchado, Emilio S., Advisory Editor, Hagras, Hani, Advisory Editor, Kóczy, László T., Advisory Editor, Kreinovich, Vladik, Advisory Editor, Lin, Chin-Teng, Advisory Editor, Lu, Jie, Advisory Editor, Melin, Patricia, Advisory Editor, Nedjah, Nadia, Advisory Editor, Nguyen, Ngoc Thanh, Advisory Editor, Wang, Jun, Advisory Editor, Hu, Zhengbing, editor, Petoukhov, Sergey V., editor, and He, Matthew, editor
- Published
- 2020
- Full Text
- View/download PDF
14. On the zero entries in a unitary matrix.
- Author
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Song, Zhiwei and Chen, Lin
- Subjects
- *
ZERO (The number) , *MATRICES (Mathematics) , *PROBLEM solving - Abstract
We investigate the number of zero entries in a unitary matrix. We show that the sets of numbers of zero entries for n × n unitary and orthogonal matrices are the same. They are both the set { 0 , 1 , ... , n 2 − n − 4 , n 2 − n − 2 , n 2 − n } for n>4. We explicitly construct examples of orthogonal matrices with the numbers in the set. We apply our results to construct a necessary condition by which a multipartite unitary operation is a product operation. The latter is a fundamental problem in quantum information. We also construct an n × n orthogonal matrix of Schmidt rank n 2 − 1 with many zero entries, and it solves an open problem in Muller-Hermes and Nechita [Operator Schmidt ranks of bipartite unitary matrices. Linear Algebra Appl. 2018;557:174—187]. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
15. 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
16. An Unusual Criterion for Normality of Nonsingular Matrices.
- Author
-
Ikramov, Kh. D.
- Abstract
The following proposition is proved: A nonsingular matrix is normal if and only if its cosquare is a unitary matrix. An unusual feature of this criterion is that normality, the most important concept in the theory of similarity transformations, is characterized in terms of transformations of an entirely different type, namely, congruence transformations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
17. A numerical range characterization of unitary matrices over a finite field.
- Author
-
Ballico, E.
- Subjects
FINITE fields ,MULTILINEAR algebra ,COMPLEX matrices ,MATRICES (Mathematics) ,UNITARY groups - Abstract
Let q be a prime power, q ≠ 2 , 3 , and let A be an n × n matrix over q 2 . We prove that the numerical range of A B and B A is the same for every rank 1 matrix B if and only if A is a multiple of a unitary matrix. The corresponding result for complex matrices was proved by Chien, Gau, Li, Tsai and Wang [Products of operators and numerical range, Linear Multilinear Algebra 64(1) (2016) 58–67]. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
18. A Survey of the Marcus–de Oliveira Conjecture
- Author
-
Huang, Huajun, Feldvoss, Jörg, editor, Grimley, Lauren, editor, Lewis, Drew, editor, Pavelescu, Andrei, editor, and Pillen, Cornelius, editor
- Published
- 2019
- Full Text
- View/download PDF
19. New Theory of Laser. Method of Density-Matrix
- Author
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Bondarev, Boris V., Parinov, Ivan A., editor, Chang, Shun-Hsyung, editor, and Kim, Yun-Hae, editor
- Published
- 2019
- Full Text
- View/download PDF
20. Polar-Precoding: A Unitary Finite-Feedback Transmit Precoder for Polar-Coded MIMO Systems.
- Author
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Piao, Jinnan, Niu, Kai, Dai, Jincheng, and Hanzo, Lajos
- Subjects
- *
MIMO systems , *DISCRETE Fourier transforms , *PHASE shift keying , *PSYCHOLOGICAL feedback - Abstract
We propose a unitary precoding scheme, namely polar-precoding, to improve the performance of polar-coded MIMO systems. In contrast to the traditional design of MIMO precoding criteria, the proposed polar-precoding scheme relies on the polarization criterion. In particular, the precoding matrix design comprises two steps. After selecting a basic matrix for maximizing the capacity in the first step, we design a unitary matrix for maximizing the polarization effect among the data streams without degrading the capacity. Our simulation results show that the proposed polar-precoding scheme outperforms the state-of-the-art DFT precoding scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
21. Advances in Evolutionary Optimization of Quantum Operators
- Author
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Petr Žufan and Michal Bidlo
- Subjects
Evolution Strategy ,Differential Evolution ,Self-adaptation of Control Parameters ,Quantum Operator ,Unitary Matrix ,Electronic computers. Computer science ,QA75.5-76.95 - Abstract
A comparative study is presented regarding the evolutionary design of quantum operators in the form of unitary matrices.A comparative study is presented regarding the evolutionary design of quantum operators in the form of unitary matrices. Three existing techniques (representations) which allow generating unitary matrices are used in various evolutionary algorithms in order to optimize their coefficients. The objective is to obtain as precise quantum operators (the resulting unitary matrices) as possible for given quantum transformations. Ordinary evolution strategy, self-adaptive evolution strategy and differential evolution are applied with various settings as the optimization algorithms for the quantum operators. These algorithms are evaluated on the tasks of designing quantum operators for the 3-qubit and 4-qubit maximum amplitude detector and a solver of a logic function of three variables in conjunctive normal form. These tasks require unitary matrices of various sizes. It will be demonstrated that the self-adaptive evolution strategy and differential evolution are able to produce remarkably better results than the ordinary evolution strategy. Moreover, the results can be improved by selecting a proper settings for the evolution as presented by a comparative evaluation.
- Published
- 2021
22. A note on completion to the unitary matrices.
- Author
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Bezerra, Johanns de Andrade
- Subjects
- *
COMPLEX matrices , *MATRICES (Mathematics) - Abstract
Let U = ( U 1 U 2 U 3 U 4 ) be a complex matrix of order n. In this paper, once a square submatrix U 1 is fixed, we present several necessary conditions on U 1 , U 2 , U 3 and U 4 , where U is a unitary matrix. Particularly, given a unitary matrix U = ( U 1 U 2 U 3 U 4 ) , we characterize the completions ( U 1 X Y U 4 ) and ( U 1 X Y Z ) to some unitary matrix, whenever U 1 and U 4 are partial isometries. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
23. Sampling the eigenvalues of random orthogonal and unitary matrices.
- Author
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Fasi, Massimiliano and Robol, Leonardo
- Subjects
- *
HAAR integral , *UNITARY groups , *RANDOM matrices , *STATISTICAL sampling , *MATRICES (Mathematics) , *COMPLEX numbers - Abstract
We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitary groups and the conditional probability distribution obtained by requiring the determinant of the sampled matrix be a given complex number on the complex unit circle. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
24. Generalizing the Butterfly Structure of the FFT
- Author
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Polcari, John, Toni, Bourama, Series Editor, and Ruffa, Anthony A., editor
- Published
- 2018
- Full Text
- View/download PDF
25. 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
26. The decomposition of an arbitrary 2w × 2w unitary matrix into signed permutation matrices.
- Author
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De Vos, Alexis and De Baerdemacker, Stijn
- Subjects
- *
COMPLEX matrices , *PERMUTATIONS , *MATRICES (Mathematics) , *STOCHASTIC matrices - Abstract
Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of 2 (say 2 w) deserve special attention, as they represent quantum qubit circuits. We investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix. It turns out to be a matrix group isomorphic to the extraspecial group E 2 2 w + 1 + of order 2 2 w + 1. An associated projective group of order 2 2 w equally suffices. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
27. A Signature Scheme on p2−dimensional Quantum System.
- Author
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Ren, Yan, Guan, Haipeng, Zhao, Qiuxia, and Ke, Lishan
- Subjects
- *
UNITARY transformations , *SCALABILITY , *DIGITAL signatures - Abstract
In this paper, we construct a signature scheme by using unitary transformation of mutually unbiased bases over p2 −dimensional Quantum System. It is a two-party signature scheme. The efficiency of this scheme is better than traditional signature schemes, since there is not any interaction or secret key sharing, and without any secure third party. The security analysis shows that our scheme is unforgeability and undeniability.. Using this method, we give an example for single-particle. The result shows that our scheme more scalability. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
28. Complex Form of Hooke's Law of Anisotropic Elastic Body.
- Author
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Martynov, N. I.
- Abstract
A complex form of Hooke's law for an anisotropic body is given, which made it possible to write down the previously known relations obtained in the simplest way. The structure of the matrix of elastic parameters and six linear invariants, which play a key role both in the connection of the stress-strain state and in the structure of the matrix of elastic parameters, have been determined. It is shown that by a certain six-dimensional unitary transformation, constructed on the basis of Hausholder matrices, the matrix of elastic moduli is reduced to the canonical form, in which the elastic moduli are invariants. Some questions of the classification of elastic materials are discussed. Formulas for the transformation of elastic moduli under three-dimensional rotation are given, as well as examples of anisotropic materials with certain properties. The possibility of constructing six invariant forms of Hooke's law is shown. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
29. Reality-based algebras with a two-dimensional representation.
- Author
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Blau, Harvey I.
- Subjects
- *
REPRESENTATIONS of algebras , *ALGEBRA - Abstract
It is shown that a two-dimensional irreducible representation of a reality-based algebra with a standard degree map is determined by the other irreducible representations of the algebra and the action of its given anti-automorphism on the distinguished basis. This generalizes known results for algebras of dimensions 5, 6, 7, and 8. Explicit conclusions equivalent to those for dimensions 5 and 6 are recovered here by specializing the procedure of proof of the main theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
30. An Extended Approach for Generating Unitary Matrices for Quantum Circuits.
- Author
-
Zhiqiang Li, Wei Zhang, Gaoman Zhang, Juan Dai, Jiajia Hu, Perkowski, Marek, and Xiaoyu Song
- Subjects
QUANTUM logic ,QUANTUM gates ,LOGIC circuits ,UNITARY operators ,SUPERCONDUCTING circuits - Abstract
In this paper, we do research on generating unitary matrices for quantum circuits automatically. We consider that quantum circuits are divided into six types, and the unitary operator expressions for each type are offered. Based on this, we propose an algorithm for computing the circuit unitary matrices in detail. Then, for quantum logic circuits composed of quantum logic gates, a faster method to compute unitary matrices of quantum circuits with truth table is introduced as a supplement. Finally, we apply the proposed algorithm to different reversible benchmark circuits based on NCT library (including NOT gate, Controlled-NOT gate, Toffoli gate) and generalized Toffoli (GT) library and provide our experimental results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
31. The Birkhoff theorem for unitary matrices of prime-power dimension.
- Author
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De Vos, Alexis and De Baerdemacker, Stijn
- Subjects
- *
MATRICES (Mathematics) , *PERMUTATION groups , *PERMUTATIONS , *DIMENSIONS - Abstract
The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity. If the dimension n of the unitary matrix equals a power of a prime p , i.e. if n = p w , then the Birkhoff decomposition does not need all n ! possible permutation matrices, as the epicirculant permutation matrices suffice. This group of permutation matrices is isomorphic to the general affine group GA(w , p) of order only p w (p w − 1) (p w − p)... (p w − p w − 1) ≪ (p w) !. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
32. Fast computation of error bounds for all eigenpairs of a Hermitian and all singular pairs of a rectangular matrix with emphasis on eigen- and singular value clusters.
- Author
-
Rump, Siegfried M. and Lange, Marko
- Subjects
- *
MATRICES (Mathematics) , *ORTHONORMAL basis , *VECTOR spaces , *COMPLEX matrices , *EIGENVALUES , *EIGENANALYSIS , *EIGENVECTORS - Abstract
We present verification methods to compute error bounds for all eigenvectors of a Hermitian matrix as well as for all singular vectors of a rectangular real or complex matrix. In case of clusters these are bounds for an orthonormal basis of the invariant subspace or singular vector space, respectively. Individual error bounds for all eigenvalues and singular values including clustered and/or multiple ones are computed as well. The computed bounds do contain the true result with mathematical certainty, and the algorithms apply to interval data as well. In that case the computed bounds are true for every real/complex matrix within the tolerances. The computational complexity to compute inclusions of all eigen/singular pairs of an n × n matrix or m × n matrix is O (n 3) or O (m n 2) operations, respectively. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. Hermitian matrices over K-octonions and their diagonalizations.
- Author
-
Golasiński, Marek and Gómez Ruiz, Francisco
- Abstract
The paper develops techniques to present an explicit diagonalization of 3 × 3 -Hermitian matrices over K -octonions O (K) via the K -exceptional group F 4 (K) provided K is a real closed field. Then, an action of F 4 (K) on the K -Cayley plane O (K) P 2 is studied to show a bijection F 4 (K) / Spin (9 , K) ⟶ ≈ O (K) P 2 for the stabilizer Spin (9 , K) of the matrix E 11 ∈ O (K) P 2 provided K is a formally real Pythagorean field. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. 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
35. Explicit Methods
- Author
-
Eidelman, Yuli, Gohberg, Israel, Haimovici, Iulian, Ball, Joseph A., Series editor, Dym, Harry, Series editor, Kaashoek, Marinus A., Series editor, Langer, Heinz, Series editor, Tretter, Christiane, Series editor, Eidelman, Yuli, Gohberg, Israel, and Haimovici, Iulian
- Published
- 2014
- Full Text
- View/download PDF
36. The matrix-valued numerical range over finite fields.
- Author
-
BALLICO, Edoardo
- Subjects
- *
FINITE fields , *FROBENIUS groups , *MATRICES (Mathematics) - Abstract
In this paper we define and study the matrix-valued k x k numerical range of n x n matrices using the Hermitian product and the product with n x k unitary matrices U (on the right with U, on the left with its adjoint U† = U-1). For all i, j = 1, ..., k we study the possible (i, j)-entries of these k x k matrices. Our results are for the case in which the base field is finite, but the same definition works over C. Instead of the degree 2 extension R → C we use the degree 2 extension Fq → Fq2, q a prime power, with the Frobenius map t → tq as the nonzero element of its Galois group. The diagonal entries of the matrix numerical ranges are the scalar numerical ranges, while often the nondiagonal entries are the entire Fq2 . We also define the matrix-valued numerical range map. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
37. CMMSE algorithms for constructing doubly stochastic matrices with the relative gain array (combined matrix) A∘A-T.
- Author
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Fuster, Robert, Gassó, Maria T., and Giménez, Isabel
- Subjects
- *
STOCHASTIC matrices , *HADAMARD matrices , *LINEAR algebra , *CHEMICAL processes , *ALGORITHMS , *MATRICES (Mathematics) , *CHEMICAL engineers - Abstract
The Combined matrix of a nonsingular matrix A is defined by ϕ (A) = A ∘ A - 1 T where ∘ means the Hadamard (entrywise) product. If the matrix A describes the relation between inputs and outputs in a multivariable process control, ϕ (A) describes the "relative gain array" (RGA) of the process and it defines the Bristol method (IEEE Trans Autom Control 1:133–134, 1966) often used for Chemical processes (McAvoy in Interaction analysis: principles and applications. Instrument Society of America, Pittsburgh, 1983; Papadourakis et al. in Ind Eng Chem Res 26(6):1259–1262, 1987; Wang et al. in Chem Eng Technol, 10.1002/ceat.201500202, 2016; Kariwala et al. in Ind Eng Chem Res 45(5):1751–1757, 10.1021/ie050790r, 2006; Golender et al. in J Chem Inf Comput Sci 21(4):196-204, 10.1021/ci00032a004, 1981). The combined matrix has been studied in several works such as Bru et al. (J Appl Math, 10.1155/2014/182354, 2014), Fiedler and Markham (Linear Algebra Appl 435:1945–1955, 2011) and Johnson and Shapiro (SIAM J Algebraic Discrete Methods 7:627–644, 1986). Since ϕ (A) = (c ij) has the property of ∑ k c ik = ∑ k c kj = 1 , ∀ i , j , when ϕ (A) ≥ 0 , ϕ (A) is a doubly stochastic matrix. In certain chemical engineering applications a diagonal of the RGA in wchich the entries are near 1 is used to determine the pairing of inputs and outputs for further design analysis. Applications of these matrices can be found in Communication Theory, related with the satellite-switched time division multiple-access systems, and about a doubly stochastic automorphism of a graph. In this paper we present new algorithms to generate doubly stochastic matrices with the Combined matrix using Hessenberg matrices in Sect. 3 and orthogonal/unitary matrices in Sect. 4. In addition, we discuss what kind of doubly stochastic matrices are obtained with our algorithms and the possibility of generating a particular doubly stochastic matrix by the map ϕ . [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
38. A New Quantum Secret Sharing Scheme Based on Mutually Unbiased Bases.
- Author
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Hao, Na, Li, Zhi-Hui, Bai, Hai-Yan, and Bai, Chen-Ming
- Subjects
- *
QUANTUM computing , *PRIME numbers , *UNITARY transformations , *ODD numbers , *QUANTUM mechanics , *COMPUTER security , *QUANTUM noise - Abstract
In this paper, we put forward a new secret sharing scheme. First, we give the mutually unbiased bases on the p2-dimensional quantum system where p is an odd prime number, and then we construct the corresponding unitary transformation based on the properties of these mutually unbiased bases. Second, we construct a (N, N) threshold secret sharing scheme by using unitary transformation between these mutually unbiased bases. At last, we analyze the scheme's security by several ways, for example, intercept-and-resend attack, entangle-and-measure attack, trojan horse attack, and so on. Using our method, we construct a single-particle quantum protocol involving only one qudit, and the method shows much more scalability than other schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
39. Defect and Equivalence of Unitary Matrices. The Fourier Case. Part II.
- Author
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Tadej, Wojciech
- Subjects
PERMUTATIONS ,MATHEMATICAL equivalence ,REPRESENTATIONS of groups (Algebra) ,MATRICES (Mathematics) ,HADAMARD matrices ,KRONECKER products - Published
- 2019
- Full Text
- View/download PDF
40. Quantum Security Computation on Shared Secrets.
- Author
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Bai, Hai-Yan, Li, Zhi-Hui, and Hao, Na
- Subjects
- *
QUANTUM computers , *CLIFFORD algebras , *QUANTUM numbers , *QUANTUM information theory , *QUANTUM computing - Abstract
Ouyang et al. proposed an (n, n) threshold quantum secret sharing scheme, where the number of participants was limited to n = 4k + 1, k ∈ Z+, and the security evaluation of the scheme was carried out accordingly. In this paper, we introduce an (n, n) threshold quantum secret sharing scheme for the number of participants n in any case (n ∈ Z+ ). The scheme is based on a quantum circuit, which consists of Clifford group gates and Toffoli gates. We study the properties of the quantum circuit in this paper and use the quantum circuit to analyze the security of the scheme for dishonest participant attack. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
41. Statistics and characterization of matrices by determinant and trace.
- Author
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Alkan, Emre and Yörük, Ekin Sıla
- Abstract
Answering a question of Erdös, Komlós proved in 1968 that almost all n×n Bernoulli matrices are nonsingular as n→∞. In this paper, we offer a new perspective on the question of Erdös by studying n×n matrices with prime number entries in an almost all sense. Precisely, it is shown that, as x→∞, the probability of randomly choosing a nonsingular n×n matrix among all n×n matrices with prime number entries that are ≤x is 1. If A is a unitary matrix, then it is well known that |detA|=1. However, the converse is far from being true. As a remedy of this defect, we search for necessary and sufficient conditions for being a unitary matrix by teaming up determinant with trace. In this way, we are led to simple characterizations of unitary matrices in the set of normal matrices. The question of which nonsingular commuting complex matrices with real eigenvalues have the same characteristic polynomial is formulated via determinant and trace conditions. Finally, through a study of eigenvectors, we obtain new characterizations of Hermitian and normal matrices. Our approach to proving these results benefits from a modular interpretation of nonsingularity and the spectral theorem for normal operators together with equality cases of classical inequalities such as the arithmetic-geometric mean inequality and the Cauchy-Schwarz inequality. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
42. Universality of Weyl unitaries
- Author
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Oluwatobi Ruth Ojo, Douglas Farenick, and Sarah Plosker
- Subjects
Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,Pauli matrices ,Mathematics::Operator Algebras ,Root of unity ,010102 general mathematics ,Identity matrix ,010103 numerical & computational mathematics ,Unitary matrix ,01 natural sciences ,Linear map ,Matrix (mathematics) ,symbols.namesake ,symbols ,Discrete Mathematics and Combinatorics ,Order (group theory) ,Geometry and Topology ,0101 mathematics ,Group theory ,Mathematics - Abstract
Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p × p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v , satisfy u p = v p = 1 p (the p × p identity matrix) and the commutation relation u v = ζ v u , where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d × d unitary matrices such that u p = v p = 1 d and u v = ζ v u , then there exists a unital completely positive linear map ϕ : M p ( C ) → M d ( C ) such that ϕ ( u ) = u and ϕ ( v ) = v . We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails. There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63] . This standard construction is generalised herein to the case p ≥ 3 , producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.
- Published
- 2022
43. Defect and Equivalence of Unitary Matrices. The Fourier Case. Part I.
- Author
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Tadej, Wojciech
- Subjects
KRONECKER products ,TENSOR products ,ABELIAN groups ,HADAMARD matrices ,HYPERPLANES - Published
- 2018
- Full Text
- View/download PDF
44. On injectivity of quantum finite automata
- Author
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Paul C. Bell and Mika Hirvensalo
- Subjects
QA75 ,FOS: Computer and information sciences ,Rational number ,Pure mathematics ,General Computer Science ,Formal Languages and Automata Theory (cs.FL) ,Computer Networks and Communications ,Computer Science - Formal Languages and Automata Theory ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Measure (mathematics) ,Theoretical Computer Science ,020204 information systems ,0202 electrical engineering, electronic engineering, information engineering ,Quantum finite automata ,Algebraic number ,QA ,F.1.1 ,F.4.1 ,F.4.3 ,Mathematics ,T1 ,Applied Mathematics ,State vector ,Unitary matrix ,Injective function ,Undecidable problem ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Computer Science::Formal Languages and Automata Theory - Abstract
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the injectivity problem of determining if the acceptance probability function of a MO-QFA is injective over all input words, i.e., giving a distinct probability for each input word. We show that the injectivity problem is undecidable for 8 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial state vector is real algebraic. We also show undecidability of this problem when the initial vector is rational, although with a huge increase in the number of states. We utilize properties of quaternions, free rotation groups, representations of tuples of rationals as linear sums of radicals and a reduction of the mixed modification of Post's correspondence problem, as well as a new result on rational polynomial packing functions which may be of independent interest., Comment: Accepted journal version (with change of name from Acceptance Ambiguity for Quantum Automata)
- Published
- 2021
45. Edge Behavior of Two-Dimensional Coulomb Gases Near a Hard Wall
- Author
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Seong-Mi Seo
- Subjects
Physics ,Nuclear and High Energy Physics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Context (language use) ,Mathematical Physics (math-ph) ,Unitary matrix ,Edge (geometry) ,Local statistics ,Measure (mathematics) ,Constraint (information theory) ,Classical mechanics ,Singular component ,Coulomb ,Mathematical Physics - Abstract
We consider a two-dimensional Coulomb gas confined to a disk when the external potential is radially symmetric. In the presence of a hard-wall constraint effective to change the equilibrium, the density of the equilibrium measure acquires a singular component at the hard wall. In the determinantal case, we study the local statistics of Coulomb particles at the hard wall and prove that their local correlations are expressed in terms of "Laplace-type" integrals, which appear in the context of truncated unitary matrices in the regime of weak non-unitarity.
- Published
- 2021
46. Quantum Computer Network Model for a Decision Making Algorithm
- Author
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Wiśniewska, Joanna, Kwiecień, Andrzej, editor, Gaj, Piotr, editor, and Stera, Piotr, editor
- Published
- 2012
- Full Text
- View/download PDF
47. Enhanced SAFER+ Algorithm for Bluetooth to Withstand Against Key Pairing Attack
- Author
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Chaudhari, Payal, Diwanji, Hiteishi, Meghanathan, Natarajan, editor, Nagamalai, Dhinaharan, editor, and Chaki, Nabendu, editor
- Published
- 2012
- Full Text
- View/download PDF
48. Generators and Relations for Un(Z[1/2,i])
- Author
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Peter Selinger and Xiaoning Bian
- Subjects
FOS: Computer and information sciences ,Physics ,Discrete mathematics ,Quantum Physics ,Computer Science - Logic in Computer Science ,Ring (mathematics) ,Group (mathematics) ,Computer Science - Emerging Technologies ,FOS: Physical sciences ,Unitary matrix ,Subring ,Logic in Computer Science (cs.LO) ,Quantum circuit ,Matrix (mathematics) ,Emerging Technologies (cs.ET) ,Computer Science::Emerging Technologies ,Quantum Physics (quant-ph) ,Complex number ,Quantum computer - Abstract
Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, omega^dagger H, and S. All of these gates have matrix entries in the ring Z[1/2,i], the smallest subring of the complex numbers containing 1/2 and i. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in Z[1/2,i] can be realized by a quantum circuit over the above gate set using at most one ancilla. In this paper, we give a finite presentation by generators and relations of U_n(Z[1/2,i]), the group of unitary nxn-matrices with entries in Z[1/2,i]., Comment: In Proceedings QPL 2021, arXiv:2109.04886
- Published
- 2021
49. An alternative canonical form for quaternionic H-unitary matrices
- Author
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D.B. Janse van Rensburg, G. J. Groenewald, André C. M. Ran, and Mathematics
- Subjects
Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,H-unitary ,010102 general mathematics ,Field (mathematics) ,010103 numerical & computational mathematics ,Unitary matrix ,01 natural sciences ,Canonical forms ,law.invention ,Matrix (mathematics) ,Invertible matrix ,law ,Skew-field of quaternions ,Linear algebra ,Discrete Mathematics and Combinatorics ,Development (differential geometry) ,Canonical form ,Geometry and Topology ,0101 mathematics ,Quaternion ,Mathematics - Abstract
The field of linear algebra over the quaternions is a research area which is still in development. In this paper we continue our research on canonical forms for a matrix pair ( A , H ) , where the matrix A is H-unitary, H is invertible and with A as well as H quaternionic matrices. We seek an invertible matrix S such that the transformations from ( A , H ) to ( S − 1 A S , S ⁎ H S ) brings the matrix A in Jordan form and simultaneously brings H into a canonical form. Canonical forms for such pairs of matrices already exist in the literature, the goal of the present paper is to add one more canonical form which specifically keeps A in Jordan form, in contrast to the existing canonical forms.
- Published
- 2021
50. Novel Statistical Wideband MIMO V2V Channel Modeling Using Unitary Matrix Transformation Algorithm
- Author
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Liang Wu, Hao Jiang, Baiping Xiong, Jian Dang, Hongming Zhang, Zaichen Zhang, and Jiangfan Zhang
- Subjects
Computer science ,Applied Mathematics ,Transmitter ,MIMO ,Unitary matrix ,Propagation delay ,Unitary transformation ,Communications system ,Computer Science Applications ,Electrical and Electronic Engineering ,Wideband ,Algorithm ,Computer Science::Information Theory ,Communication channel - Abstract
For efficiently investigating the statistical properties of wideband multiple-input multiple-output (MIMO) channels for vehicle-to-vehicle (V2V) communication scenarios, we propose a novel computationally efficient solution to estimate the parameters of the proposed channel model for different propagation delays in this paper. To be specific, we first introduce a Unitary transformation method to estimate the propagation delay of the proposed channel model for the first tap in the preliminary stage before the mobile transmitter (MT) and mobile receiver (MR) move. Then, we estimate the real-time angular parameters based on the estimated delay and moving time/directions/velocities of the MT and MR. Furthermore, we estimate the expressions of the real-time complex channel impulse responses (CIRs), which can be used to characterize the physical properties of the proposed channel model, by substituting the estimates of the time-varying AoD and AoA and model parameters into the complex CIRs. Numerical results of the channel characteristics fit the theory results very well, which validate that the proposed channel model is practical for characterizing the beyond fifth-generation (B5G) V2V communication systems.
- Published
- 2021
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