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Journal articles on the topic 'Singular M-matrix'

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Published: 10 March 2023

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1

Montaño, Emedin, Mario Salas, and Ricardo L. Soto. "Nonnegativity Preservation under Singular Values Perturbation." Mathematical Problems in Engineering 2009 (2009): 1–25. http://dx.doi.org/10.1155/2009/301582.

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We study how singular values and singular vectors of a matrixAchange, under matrix perturbations of the formA+αuivi∗andA+αupvq∗,p≠q, whereα∈ℝ,Ais anm×npositive matrix with singular valuesσ1≥σ2≥⋯≥σr>0,r=min⁡{m,n}, anduj,vk, j=1,…,m;k=1,…,n, are the left and right singular vectors, respectively. In particular we give conditions under which this kind of perturbations preserve nonnegativity and certain matrix structures.

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2

Kaper, Hans G., and Allan M. Krall. "M(λ)-computation for singular differential systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 112, no. 3-4 (1989): 327–30. http://dx.doi.org/10.1017/s0308210500018783.

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SynopsisDepending upon the initial data associated with the fundamental matrix, the function M(λ), used to generate L2-solutions of homogeneous linear differential systems, may vary. We show that there is a matrix bilinear transformation between such functions M(λ) with different initial data and illustrate how the result can be used to simplify the calculation of a specific M(λ)-function for a scalar second-order problem.

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3

Keilson, J., and L. D. Servi. "The matrix M/M/∞ system: retrial models and Markov Modulated sources." Advances in Applied Probability 25, no. 2 (June 1993): 453–71. http://dx.doi.org/10.2307/1427662.

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The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

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Keilson, J., and L. D. Servi. "The matrix M/M/∞ system: retrial models and Markov Modulated sources." Advances in Applied Probability 25, no. 02 (June 1993): 453–71. http://dx.doi.org/10.1017/s0001867800025441.

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The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

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5

Li, Ren-cang, Wei-chao Wang, and Wei-guo Wang. "Deflating irreducible singular M-matrix algebraic Riccati equations." Numerical Algebra, Control and Optimization 3, no. 3 (July 2013): 491–518. http://dx.doi.org/10.3934/naco.2013.3.491.

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6

Ding, Xiucai. "Singular vector distribution of sample covariance matrices." Advances in Applied Probability 51, no. 01 (March 2019): 236–67. http://dx.doi.org/10.1017/apr.2019.10.

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AbstractWe consider a class of sample covariance matrices of the form Q = TXX*T*, where X = (xij) is an M×N rectangular matrix consisting of independent and identically distributed entries, and T is a deterministic matrix such that T*T is diagonal. Assuming that M is comparable to N, we prove that the distribution of the components of the right singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of xij coincide with the Gaussian random variables. For the right singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of xij match those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume that T is diagonal.

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7

Bendito, E., A. Carmona, A. M. Encinas, and M. Mitjana. "The M-matrix inverse problem for singular and symmetric Jacobi matrices." Linear Algebra and its Applications 436, no. 5 (March 2012): 1090–98. http://dx.doi.org/10.1016/j.laa.2011.06.044.

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8

Cantó, Rafael, and Joan-Josep Climent. "Singular graph and extension of jordan chains of an M-Matrix." Linear Algebra and its Applications 241-243 (July 1996): 167–89. http://dx.doi.org/10.1016/0024-3795(95)00584-6.

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9

Truhar, Ninoslav, and Maja Petrač. "Damping Optimization of Linear Vibrational Systems with a Singular Mass Matrix." Mathematics 10, no. 11 (May 28, 2022): 1854. http://dx.doi.org/10.3390/math10111854.

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We present two novel results for small damped oscillations described by the vector differential equation Mx¨+Cx˙+Kx=0, where the mass matrix M can be singular, but standard deflation techniques cannot be applied. The first result is a novel formula for the solution X of the Lyapunov equation ATX+XA=−I, where A=A(v) is obtained from M,C(v)∈Rn×n, and K∈Rn×n, which are the so-called mass, damping, and stiffness matrices, respectively, and rank(M)=n−1. Here, C(v) is positive semidefinite with rank(C(v))=1. Using the obtained formula, we propose a very efficient way to compute the optimal damping matrix. The second result was obtained for a different structure, where we assume that dim(N(M))≥1 and internal damping exists (usually a small percentage of the critical damping). For this structure, we introduce a novel linearization, i.e., a novel construction of the matrix A in the Lyapunov equation ATX+XA=−I, and a novel optimization process. The proposed optimization process computes the optimal damping C(v) that minimizes a function v↦trace(ZX) (where Z is a chosen symmetric positive semidefinite matrix) using the approximation function g(v)=cv+av+bv, for the trace function f(v)≐trace(ZX(v)). Both results are illustrated with several corresponding numerical examples.

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10

Barlow, Jesse L. "On the Smallest Positive Singular Value of a Singular M-Matrix with Applications to Ergodic Markov Chains." SIAM Journal on Algebraic Discrete Methods 7, no. 3 (July 1986): 414–24. http://dx.doi.org/10.1137/0607047.

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11

Li, Wen, and Mou-Cheng Zhang. "On upper triangular block weak regular splittings of a singular M-matrix." Linear Algebra and its Applications 233 (January 1996): 175–87. http://dx.doi.org/10.1016/0024-3795(94)00068-9.

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12

Peller, V. V., and N. J. Young. "Superoptimal approximation by meromorphic functions." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 3 (April 1996): 497–511. http://dx.doi.org/10.1017/s0305004100074375.

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AbstractLet G be a matrix function of type m × n and suppose that G is expressible as the sum of an H∞ function and a continuous function on the unit circle. Suppose also that the (k – 1)th singular value of the Hankel operator with symbol G is greater than the kth singular value. Then there is a unique superoptimal approximant to G in : that is, there is a unique matrix function Q having at most k poles in the open unit disc which minimizes s∞(G – Q) or, in other words, which minimizes the sequencewith respect to the lexicographic ordering, whereand Sj(·) denotes the jth singular value of a matrix. This result is due to the present authors [PY1] in the case k = 0 (when the hypothesis on the Hankel singular values is vacuous) and to S. Treil[T2] in general. In this paper we give a proof of uniqueness by a diagonalization argument, a high level algorithm for the computation of the superoptimal approximant and a recursive parametrization of the set of all optimal solutions of a matrix Nehari—Takagi problem.

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13

Ani, Ani, Mashadi Mashadi, and Sri Gemawati. "Invers Moore-Penrose pada Matriks Turiyam Simbolik Real." Jambura Journal of Mathematics 5, no. 1 (January 18, 2023): 95–114. http://dx.doi.org/10.34312/jjom.v5i1.16304.

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The symbolic Turiyam matrix is a matrix whose entries contain symbolic Turiyam. Inverse matrices can generally be determined if the matrix is a non-singular square matrix. Currently the inverse of the symbolic Turiyam matrix of size m × n with m 6= n can be determined by the Moore-Penrose inverse. The purpose of this research is to determine the inverse Moore-Penrose algorithm on a real symbolic Turiyam matrix of size m × n with m 6= n. Algebraic operations on symbolic Turiyam is a method used to obtain the Moore-Penrose inverse on real symbolic Turiyam matrices by applying symbolic Turiyam algebraic operations on the concept of Moore-Penrose inverses. The main result obtained is the inverse Moore-Penrose algorithm on the real symbolic Turiyam matrix. The demonstration example given shows that the Moore-Penrose inverse on a real symbolic Turiyam matrix always exists even though the matrix is not a square matrix.

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14

FERREIRA, CHELO, JOSE L. LOPEZ, and ESTER PEREZ SINUSIA. "The Picard-Lindel of’s theorem at a regular singular point." Carpathian Journal of Mathematics 29, no. 2 (2013): 167–78. http://dx.doi.org/10.37193/cjm.2013.02.13.

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We consider initial value problems of the form..., where f : [−a, b] × U → Cn is a continuous function in its variables and U ⊂ Cn is an open set. D(x) is an n × n diagonal matrix whose first n − m diagonal entries are 1 and the last m diagonal entries are x, with m = 0, 1, 2, . . . or n. This is an initial value problem where the initial condition is given at a regular singular point of the system of differential equations. The main result of this paper is an existence and uniqueness theorem for the solution of this initial value problem. It is shown that this problem has a unique solution and the Picard-Lindelof’s expansion converges to that solution if the function ¨ F(y, x) := xD−1 (x)f (x, y) is Lipschitz continuous in the variables y with Lipschitz constant L of the form L = N + Mxp for a certain p > 0, M > 0 and 0 ≤ N < 1. When we add the condition y (s) ∈ C[−a, b], s ∈ N, to the formulation of the problem and the Taylor polynomial of y at x = 0 and degree s − 1 is available from the differential equation, then the same conclusion is true with a less restrictive condition upon N: 0 ≤ N < s + 1. The standard Picard-Lindelof’s ¨ theorem is the particular case of the problem studied here obtained for m = 0 (D(x) is the identity matrix), N = 0, p = 1 and M is the Lipschitz constant of f (x, y).

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15

Xu, Yong-Xian, D. Kohli, and Tzu-Chen Weng. "Direct Differential Kinematics of Hybrid-Chain Manipulators Including Singularity and Stability Analyses." Journal of Mechanical Design 116, no. 2 (June 1, 1994): 614–21. http://dx.doi.org/10.1115/1.2919422.

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A general formulation for the differential kinematics of hybrid-chain manipulators is developed based on transformation matrices. This formulation leads to velocity and acceleration analyses, as well as to the formation of Jacobians for singularity and unstable configuration analyses. A manipulator consisting of n nonsymmetrical subchains with an arbitrary arrangement of actuators in the subchain is called a hybrid-chain manipulator in this paper. The Jacobian of the manipulator (called here the system Jacobian) is a product of two matrices, namely the Jacobian of a leg and a matrix M containing the inverse of a matrix Dk, called the Jacobian of direct kinematics. The system Jacobian is singular when a leg Jacobian is singular; the resulting singularity is called the inverse kinematic singularity and it occurs at the boundary of inverse kinematic solutions. When the Dk matrix is singular, the M matrix and the system Jacobian do not exist. The singularity due to the singularity of the Dk matrix is the direct kinematic singularity and it provides positions where the manipulator as a whole loses at least one degree of freedom. Here the inputs to the manipulator become dependent on each other and are locked. While at these positions, the platform gains at least one degree of freedom, and becomes statically unstable. The system Jacobian may be used in the static force analysis. A stability index, defined in terms of the condition number of the Dk matrix, is proposed for evaluating the proximity of the configuration to the unstable configuration. Several illustrative numerical examples are presented.

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16

Bentbib, A. H., and A. Kanber. "Block Power Method for SVD Decomposition." Analele Universitatii "Ovidius" Constanta - Seria Matematica 23, no. 2 (June 1, 2015): 45–58. http://dx.doi.org/10.1515/auom-2015-0024.

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Abstract We present in this paper a new method to determine the k largest singular values and their corresponding singular vectors for real rectangular matrices A ∈ Rn×m. Our approach is based on using a block version of the Power Method to compute an k-block SV D decomposition: Ak = Uk∑kVkT , where ∑k is a diagonal matrix with the k largest non-negative, monotonically decreasing diagonal σ1≥ σ2 ⋯ ≥ σk. Uk and Vk are orthogonal matrices whose columns are the left and right singular vectors of the k largest singular values. This approach is more efficient as there is no need of calculation of all singular values. The QR method is also presented to obtain the SV D decomposition.

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17

Zhu, Huizhi, Wenxia Xu, Baocheng Yu, Feng Ding, Lei Cheng, and Jian Huang. "A Novel Hybrid Algorithm for the Forward Kinematics Problem of 6 DOF Based on Neural Networks." Sensors 22, no. 14 (July 16, 2022): 5318. http://dx.doi.org/10.3390/s22145318.

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The closed kinematic structure of Gough–Stewart platforms causes the kinematic control problem, particularly forward kinematics. In the traditional hybrid algorithm (backpropagation neural network and Newton–Raphson), it is difficult for the neural network part to train different datasets, causing training errors. Moreover, the Newton–Raphson method is unable to operate on a singular Jacobian matrix. In this study, in order to solve the forward kinematics problem of Gough–Stewart platforms, a new hybrid algorithm is proposed based on the combination of an artificial bee colony (ABC)–optimized BP neural network (ABC–BPNN) and a numerical algorithm. ABC greatly improves the prediction ability of neural networks and can provide a superb initial value to numerical algorithms. In the design of numerical algorithms, a modification of Newton’s method (QMn-M) is introduced to solve the problem that the traditional algorithm model cannot be solved when it is trapped in singular matrix. Results show that the maximal improvement in ABC–BPNN error optimization was 46.3%, while the RMSE index decreased by 42.1%. Experiments showed the feasibility of QMn-M in solving singular matrix data, while the percentage improvement in performance for the average number of iterations and required time was 14.4% and 13.9%, respectively.

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18

Zhou, Duan-Mei, Xiang-Xing Ye, Qing-Wen Wang, Jia-Wen Ding, and Wen-Yu Hu. "Explicit solutions of the Yang-Baxter-like matrix equation for a singular diagonalizable matrix with three distinct eigenvalues." Filomat 35, no. 12 (2021): 3971–82. http://dx.doi.org/10.2298/fil2112971z.

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Let A be a singular diagonalizable complex matrix with three distinct eigenvalues. We derive all explicit solutions X of the Yang-Baxter-like matrix equation AXA = XAX, by taking advantage of the Jordan form structure of A. The result generates the formula obtained in Chen et al. (2019) and M. Saeed Ibrahim Adam et al. (2019). We give examples to illustrate the validity of the results obtained in this paper.

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19

Zhou, Zheng-Da, and Shi-Cai Gong. "A Geometric Property of the Laplacian matrix of a Connected Nonsingular Mixed Graph." Journal of Chemistry 2020 (February 28, 2020): 1–4. http://dx.doi.org/10.1155/2020/6210758.

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In this paper, we give a geometric interpretation of the Laplacian matrix of a connected nonsingular mixed graph which generalizes the results of M. Fiedler (M. Fiedler, Geometry of the Laplacian, Linear Algebra Appl., 2005, 403: 409–413). In addition, the relations of geometric properties between a connected (singular or nonsingular) mixed graph, and all its resigned graphs will be characterized.

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20

Mokhtar-Kharroubi, Mustapha. "Form-perturbation theory for higher-order elliptic operators and systems by singular potentials." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2185 (October 19, 2020): 20190621. http://dx.doi.org/10.1098/rsta.2019.0621.

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We give a form-perturbation theory by singular potentials for scalar elliptic operators on L 2 ( R d ) of order 2 m with Hölder continuous coefficients. The form-bounds are obtained from an L 1 functional analytic approach which takes advantage of both the existence of m -gaussian kernel estimates and the holomorphy of the semigroup in L 1 ( R d ) . We also explore the (local) Kato class potentials in terms of (local) weak compactness properties. Finally, we extend the results to elliptic systems and singular matrix potentials. This article is part of the theme issue ‘Semigroup applications everywhere’.

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21

Ozturk, Semra. "On $m$-th roots of nilpotent matrices." Electronic Journal of Linear Algebra 37 (December 14, 2021): 718–33. http://dx.doi.org/10.13001/ela.2021.6331.

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A new necessary and sufficient condition for the existence of an $m$-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the $m$-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an $m$-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an $m$-th root is corrected and generalized. Moreover, for a singular matrix having an $m$-th root with a pair of nilpotent Jordan blocks of sizes $s$ and $l$, a new $m$-th root is constructed by replacing that pair by another one of sizes $s+i$ and $l-i$, for special $s,l,i$. This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix $A$ over an arbitrary field that is a sum of two commuting matrices, several results for the existence of $m$-th roots of $A^k$ are obtained.

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22

Saha, Manideepa, and Sriparna Bandopadhyay. "Combinatorial properties of generalized M-matrices." Electronic Journal of Linear Algebra 30 (February 8, 2015): 550–76. http://dx.doi.org/10.13001/1081-3810.1974.

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An M_â¨-matrix has the form A = sI â B with s â¥ Ï(B) and B^k is entrywise nonnegative for all sufficiently large integers k. In this paper, the existence of a preferred basis for a singular M_â¨- matrix A = sI â B with index(B) â¤ 1 is proven. Some equivalent conditions for the equality of the height and level characteristics of A are studied. The well structured property of the reduced graph of A is discussed. Also the possibility of the existence of preferred basis for another generalization of M-matrices, known as GM-matrices, is studied.

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23

Ataei, Alireza. "Improved Qrginv Algorithm for Computing Moore-Penrose Inverse Matrices." ISRN Applied Mathematics 2014 (March 12, 2014): 1–5. http://dx.doi.org/10.1155/2014/641706.

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Katsikis et al. presented a computational method in order to calculate the Moore-Penrose inverse of an arbitrary matrix (including singular and rectangular) (2011). In this paper, an improved version of this method is presented for computing the pseudo inverse of an m×n real matrix A with rank r>0. Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that obtained by Katsikis et al.

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24

Szwarc, Michael. "Solution of the equilibrium copolymerization equations for the case of the singular matrix M." Macromolecules 19, no. 12 (December 1986): 3004–6. http://dx.doi.org/10.1021/ma00166a024.

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25

Chen, Dongjun, and Yun Zhanga. "Singular value inequalities for real and imaginary parts of matrices." Filomat 30, no. 10 (2016): 2623–29. http://dx.doi.org/10.2298/fil1610623c.

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Let A = Re A + i Im A be the Cartesian decomposition of square matrix A of order n with Re A = A+A*/2 and Im A = A-A*/2i. Fan-Hoffman?s result asserts that ?j(ReA)? sj(A), j=1,..., n, where ?j(M) and sj(M) stand for the jth largest eigenvalue of M and the jth largest singular value of M, respectively. We investigate singular value inequalities for real and imaginary parts of matrices and prove the following inequalities: sj(Re A) ? 1/4 sj([(|A|+|A*|)-(A+A*)]?[(|A|+|A*|)+(A+A*)]), and sj(Im A) ? 1/4 sj([(|A|+|A*|)- i(A*-A)] ? [(|A|+|A*|) + i(A*-A)]), j = 1,..., n. In particular, we have sj(Re A) ? 1/2 sj((|A|+|A*|)? (|A|+|A*|)), and sj(Im A) ? 1/2 sj((|A|+|A*|)? (|A|+|A*|)), j=1,..., n. Moreover, we also show that these inequalities are sharp.

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26

Bierkens, Joris, and André Ran. "A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable?" Linear Algebra and its Applications 457 (September 2014): 191–208. http://dx.doi.org/10.1016/j.laa.2014.05.022.

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27

Sheng, Xingping, and Tao Wang. "An iterative method to compute Moore-Penrose inverse based on gradient maximal convergence rate." Filomat 27, no. 7 (2013): 1269–76. http://dx.doi.org/10.2298/fil1307269s.

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In this paper, we present an iterative method based on gradient maximal convergence rate to compute Moore-Penrose inverse A+ of a given matrix A. By this iterative method, when taken the initial matrix X0 = A*, the M-P inverse A+ can be obtained with maximal convergence rate in absence of round off errors. In the end, a numerical example is given to illustrate the effectiveness, accuracy and its computation time, which are all superior than the other methods for the large singular matrix.

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Kuijlaars, Arno B. J., and Dries Stivigny. "Singular values of products of random matrices and polynomial ensembles." Random Matrices: Theory and Applications 03, no. 03 (July 2014): 1450011. http://dx.doi.org/10.1142/s2010326314500117.

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Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. We also show that the squared singular values of the product of M - 1 complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and we derive a double integral representation for the correlation kernel associated with this ensemble. We use this to calculate the scaling limit at the hard edge, which turns out to be the same scaling limit as the one found by Kuijlaars and Zhang for the squared singular values of a product of M complex Ginibre matrices. Our final result is that these limiting kernels also appear as scaling limits for the biorthogonal ensembles of Borodin with parameter θ > 0, in case θ or 1/θ is an integer. This further supports the conjecture that these kernels have a universal character.

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HACHEM, WALID, PHILIPPE LOUBATON, XAVIER MESTRE, JAMAL NAJIM, and PASCAL VALLET. "LARGE INFORMATION PLUS NOISE RANDOM MATRIX MODELS AND CONSISTENT SUBSPACE ESTIMATION IN LARGE SENSOR NETWORKS." Random Matrices: Theory and Applications 01, no. 02 (April 2012): 1150006. http://dx.doi.org/10.1142/s2010326311500067.

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In array processing, a common problem is to estimate the angles of arrival of K deterministic sources impinging on an array of M antennas, from N observations of the source signal, corrupted by Gaussian noise. In the so-called subspace methods, the problem reduces to estimate a quadratic form (called "localization function") of a certain projection matrix related to the source signal empirical covariance matrix. The estimates of the angles of arrival are then obtained by taking the K deepest local minima of the estimated localization function. Recently, a new subspace estimation method has been proposed, in the context where the number of available samples N is of the same order of magnitude than the number of sensors M. In this context, the traditional subspace methods tend to fail because they are based on the empirical covariance matrix of the observations which is a poor estimate of the source signal covariance matrix. The new subspace method is based on a consistent estimator of the localization function in the regime where M and N tend to +∞ at the same rate. However, the consistency of the angles estimator was not addressed, and the purpose of this paper is to prove this consistency in the previous asymptotic regime. For this, we prove the property that the singular values of M × N Gaussian information plus noise matrix escape from certain intervals is an event of probability decreasing at rate [Formula: see text] for all p. A regularization trick is also introduced, which allows to confine these singular values into certain intervals and to use standard tools as Poincaré inequality to characterize any moments of the estimator. These results are believed to be of independent interest.

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30

Carmona, A., A. M. Encinas, and M. Mitjana. "A combinatorial expression for the group inverse of symmetric M-matrices." Special Matrices 9, no. 1 (January 1, 2021): 275–96. http://dx.doi.org/10.1515/spma-2020-0137.

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Abstract By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general symmetric M-matrices with no restrictions, this means that we do not have to assume the diagonally dominance hypothesis. We express the group inverse of a symmetric M–matrix in terms of the weight of spanning rooted forests. In fact, we give a combinatorial expression for both the determinant of the considered matrix and the determinant of any submatrix obtained by deleting a row and a column. Moreover, the singular case is obtained as a limit case when certain parameter goes to zero. In particular, we recover some known results regarding trees. As examples that illustrate our results we give the expressions for the Group inverse of any symmetric M-matrix of order two and three. We also consider the case of the cycle C 4 an example of a non-contractible situation topologically different from a tree. Finally, we obtain some relations between combinatorial numbers, such as Horadam, Fibonacci or Pell numbers and the number of spanning rooted trees on a path.

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31

Sheng, Xingping. "Execute Elementary Row and Column Operations on the Partitioned Matrix to Compute M-P InverseA†." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/596049.

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We first study the complexity of the algorithm presented in Guo and Huang (2010). After that, a new explicit formula for computational of the Moore-Penrose inverseA†of a singular or rectangular matrixA. This new approach is based on a modified Gauss-Jordan elimination process. The complexity of the new method is analyzed and presented and is found to be less computationally demanding than the one presented in Guo and Huang (2010). In the end, an illustrative example is demonstrated to explain the corresponding improvements of the algorithm.

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32

Franca, Willian, and Nelson Louza. "Generalized Commuting Maps On The Set of Singular Matrices." Electronic Journal of Linear Algebra 35 (December 5, 2019): 533–42. http://dx.doi.org/10.13001/ela.2019.5173.

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Let Mn(K) be the ring of all n × n matrices over a field K. In the present paper, additive mappings G : Mn(K) → Mn(K) such that [[G(y), y], y] = 0 for all singular matrix y will be characterized. Precisely, it will be proved that G(x) = λx + µ(x) for all x ∈ Mn(K), where λ ∈ K and µ is a central map. As an application, the description is given of all additive maps g : Mn(K) → Mn(K) such that [[g(yk1 ), yk2 ], yk3 ] = 0 for all singular matrices y ∈ Mn(K), where m ∈ N∗.

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33

Shi, Lin, Taibin Gan, Hong Zhu, and Xianming Gu. "The Exact Distribution of the Condition Number of Complex Random Matrices." Scientific World Journal 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/729839.

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LetGm×n (m≥n)be a complex random matrix andW=Gm×nHGm×nwhich is the complex Wishart matrix. Letλ1>λ2>…>λn>0andσ1>σ2>…>σn>0denote the eigenvalues of theWand singular values ofGm×n, respectively. The 2-norm condition number ofGm×nisκ2Gm×n=λ1/λn=σ1/σn. In this paper, the exact distribution of the condition number of the complex Wishart matrices is derived. The distribution is expressed in terms of complex zonal polynomials.

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34

BEHNCKE, HORST. "SPECTRAL THEORY OF HIGHER ORDER DIFFERENTIAL OPERATORS." Proceedings of the London Mathematical Society 92, no. 1 (December 19, 2005): 139–60. http://dx.doi.org/10.1017/s0024611505015480.

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The absolutely continuous spectrum of a very general class of differential operators of order $2n$ is determined, for operators whose coefficients satisfy conditions that combine smoothness and decay properties. The main methods are asymptotic integration and the analysis of the associated $M$-matrix. The form of the solutions precludes the absence of a singular continuous spectrum.

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35

GLÄSSNER, UWE, STEPHAN GÜSKEN, THOMAS LIPPERT, GERO RITZENHÖFER, KLAUS SCHILLING, and ANDREAS FROMMER. "HOW TO COMPUTE GREEN'S FUNCTIONS FOR ENTIRE MASS TRAJECTORIES WITHIN KRYLOV SOLVERS." International Journal of Modern Physics C 07, no. 05 (October 1996): 635–44. http://dx.doi.org/10.1142/s0129183196000533.

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The availability of efficient Krylov subspace solvers plays a vital role in the solution of a variety of numerical problems in computational science. Here we consider lattice field theory. We present a new general numerical method to compute many Green's functions for complex non-singular matrices within one iteration process. Our procedure applies to matrices of structure A = D − m, with m proportional to the unit matrix, and can be integrated within any Krylov subspace solver. We can compute the derivatives x(n) of the solution vector x with respect to the parameter m and construct the Taylor expansion of x around m. We demonstrate the advantages of our method using a minimal residual solver. Here the procedure requires one intermediate vector for each Green's function to compute. As real-life example, we determine a mass trajectory of the Wilson fermion matrix for lattice QCD. Here we find that we can obtain Green's functions at all masses ≥ m at the price of one inversion at mass m.

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Klymchuk, Tetiana. "Regularizing algorithm for mixed matrix pencils." Applied Mathematics and Nonlinear Sciences 2, no. 1 (April 18, 2017): 123–30. http://dx.doi.org/10.21042/amns.2017.1.00010.

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AbstractP. Van Dooren (1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm to square complex matrices with respect to consimilarity transformations $\begin{array}{} \displaystyle A \mapsto SA{\bar S^{ - 1}} \end{array}$ and to pairs of m × n complex matrices with respect to transformations $\begin{array}{} \displaystyle (A,B) \mapsto (SAR,SB\bar R) \end{array}$, in which S and R are nonsingular matrices.

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37

Barreiro, Andrea, Jared Bronski, and Paul K. Newton. "Spectral gradient flow and equilibrium configurations of point vortices." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2118 (December 23, 2009): 1687–702. http://dx.doi.org/10.1098/rspa.2009.0419.

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We formulate the problem of finding equilibrium configurations of N -point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew-symmetric matrix M whose nullspace structure determines the (real) strengths, rotational frequency and translational velocity of the configuration. A generic configuration gives rise to a matrix with empty nullspace, and hence is not a relative equilibrium for any choice of vortex strengths. We formulate the problem as a gradient flow in the space of square covariance matrices M T M . The evolution equation for drives the configuration to one with a real nullspace, establishing the existence of an equilibrium for vortex strengths that are elements of the nullspace of the matrix. We formulate both the unconstrained gradient flow problem where the point vortex strengths are determined a posteriori by the nullspace of M and the constrained problem where the point vortex strengths are chosen a priori and one seeks configurations for which those strengths are elements of the nullspace.

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38

Dong, Liqiang, Jicheng Li, and Guo Li. "The double deflating technique for irreducible singular M-matrix algebraic Riccati equations in the critical case." Linear and Multilinear Algebra 67, no. 8 (April 29, 2018): 1653–84. http://dx.doi.org/10.1080/03081087.2018.1466862.

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Huang, Tsung-Ming, Wei-Qiang Huang, Ren-Cang Li, and Wen-Wei Lin. "A new two-phase structure-preserving doubling algorithm for critically singular M -matrix algebraic Riccati equations." Numerical Linear Algebra with Applications 23, no. 2 (November 5, 2015): 291–313. http://dx.doi.org/10.1002/nla.2025.

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40

GÖTZE, JÜRGEN. "EFFICIENT PARALLEL IMPLEMENTATION OF KOGBETLIANTZ'S SVD ALGORITHM." International Journal of High Speed Electronics and Systems 04, no. 01 (March 1993): 35–53. http://dx.doi.org/10.1142/s0129156493000030.

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In this paper efficient implementations of Kogbetliantz's algorithm for computing the singular value decomposition (SVD) of an m × n (m ≥ n) matrix on an upper triangular array of processors are presented. Modifications of the rotation evaluations, as approximations and factorizations of the rotations, are considered in order to obtain a reduction of the hardware and time requirements of the processor array. Particularly, it is shown that only the combination of these modifications enables the derivation of square root free or square root and division free algorithms.

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41

Clemence, Dominic P. "On the Singular Behaviour of the Titchmarsh-Weyl m-Function for the Perturbed Hill’s Equation on the Line." Canadian Mathematical Bulletin 40, no. 4 (December 1, 1997): 416–21. http://dx.doi.org/10.4153/cmb-1997-049-x.

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AbstractFor the perturbed Hill’s equation on the line,we study the behaviour of the matrix m-function at the spectral gap endpoints. In particular, we extend the result of Hinton, Klaus and Shaw that En, a gap endpoint, is a half-bound state (HBS) if and only if (E − En)½m(E) approaches a nonzero constant as E → En, to the present case.

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42

Albreem, Mahmoud A., Mohammed H. Alsharif, and Sunghwan Kim. "A Low Complexity Near-Optimal Iterative Linear Detector for Massive MIMO in Realistic Radio Channels of 5G Communication Systems." Entropy 22, no. 4 (March 28, 2020): 388. http://dx.doi.org/10.3390/e22040388.

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Massive multiple-input multiple-output (M-MIMO) is a substantial pillar in fifth generation (5G) mobile communication systems. Although the maximum likelihood (ML) detector attains the optimum performance, it has an exponential complexity. Linear detectors are one of the substitutions and they are comparatively simple to implement. Unfortunately, they sustain a considerable performance loss in high loaded systems. They also include a matrix inversion which is not hardware-friendly. In addition, if the channel matrix is singular or nearly singular, the system will be classified as an ill-conditioned and hence, the signal cannot be equalized. To defeat the inherent noise enhancement, iterative matrix inversion methods are used in the detectors’ design where approximate matrix inversion is replacing the exact computation. In this paper, we study a linear detector based on iterative matrix inversion methods in realistic radio channels called QUAsi Deterministic RadIo channel GenerAtor (QuaDRiGa) package. Numerical results illustrate that the conjugate-gradient (CG) method is numerically robust and obtains the best performance with lowest number of multiplications. In the QuaDRiGA environment, iterative methods crave large n to obtain a pleasurable performance. This paper also shows that when the ratio between the user antennas and base station (BS) antennas ( β ) is close to 1, iterative matrix inversion methods are not attaining a good detector’s performance.

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Sauter, Stefan A. "On the stability of the incomplete Cholesky decomposition for a singular perturbed problem, where the coefficient matrix is not an M-matrix." Numerical Linear Algebra with Applications 2, no. 1 (January 1995): 17–28. http://dx.doi.org/10.1002/nla.1680020103.

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44

Kumar, Shubham, and Deepmala. "A note on the unique solvability condition for generalized absolute value matrix equation." Journal of Numerical Analysis and Approximation Theory 51, no. 1 (September 17, 2022): 83–87. http://dx.doi.org/10.33993/jnaat511-1263.

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The spectral radius condition \[\rho (\vert A^{-1} \vert\cdot \vert B \vert)<1\]for the unique solvability of generalized absolute value matrix equation (GAVME) \[AX + B \vert X \vert = D\] is provided. For some instances, our condition is superior to the earlier published singular values conditions \(\sigma_{\max}(\vert B \vert)<\sigma_{\min}(A)\) [M. Dehghan, 2020] and \(\sigma_{\max}(B)<\sigma_{\min}(A)\) [Kai Xie, 2021]. For the validity of our condition, we also provided an example.

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45

Dhrymes, Phoebus J. "Autoregressive Errors in Singular Systems of Equations." Econometric Theory 10, no. 2 (June 1994): 254–85. http://dx.doi.org/10.1017/s0266466600008410.

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We consider a system of m general linear models, where the system error vector has a singular covariance matrix owing to various “adding up” requirements and, in addition, the error vector obeys an autoregressive scheme. The paper reformulates the problem considered earlier by Berndt and Savin [8] (BS), as well as others before them; the solution, thus obtained, is far simpler, being the natural extension of a restricted least-squares-like procedure to a system of equations. This reformulation enables us to treat all parameters symmetrically, and discloses a set of conditions which is different from, and much less stringent than, that exhibited in the framework provided by BS.Finally, various extensions are discussed to (a) the case where the errors obey a stable autoregression scheme of order r; (b) the case where the errors obey a moving average scheme of order r; (c) the case of “dynamic” vector distributed lag models, that is, the case where the model is formulated as autoregressive (in the dependent variables), and moving average (in the explanatory variables), and the errors are specified to be i.i.d.

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Stevanovic, Dragan, Abreu de, Freitas de, Cybele Vinagre, and Renata Del-Vecchio. "On the oriented incidence energy and decomposable graphs." Filomat 23, no. 3 (2009): 243–49. http://dx.doi.org/10.2298/fil0903243s.

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Let G be a simple graph with n vertices and m edges. Let edges of G be given an arbitrary orientation, and let Q be the vertex-edge incidence matrix of such oriented graph. The oriented incidence energy of G is then the sum of singular values of Q. We show that for any n 2 N, there exists a set of n graphs with O(n) vertices having equal oriented incidence energy.

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47

Stevanovic, Dragan. "Oriented incidence energy and threshold graphs." Filomat 25, no. 2 (2011): 1–8. http://dx.doi.org/10.2298/fil1102001s.

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Let G be a simple graph with n vertices and m edges. Let edges of G be given an arbitrary orientation, and let Q be the vertex-edge incidence matrix of such oriented graph. The oriented incidence energy of G is then the sum of singular values of Q. We show that for any n?9, there exists at least ([n/9]/2)+1 distinct pairs of graphs on n vertices having equal oriented incidence energy.

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48

Romanov, Elad, and Matan Gavish. "Near-optimal matrix recovery from random linear measurements." Proceedings of the National Academy of Sciences 115, no. 28 (June 25, 2018): 7200–7205. http://dx.doi.org/10.1073/pnas.1705490115.

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In matrix recovery from random linear measurements, one is interested in recovering an unknown M-by-N matrixX0fromn<MNmeasurementsyi=Tr(Ai⊤X0), where eachAiis an M-by-N measurement matrix with i.i.d. random entries,i=1,…,n. We present a matrix recovery algorithm, based on approximate message passing, which iteratively applies an optimal singular-value shrinker—a nonconvex nonlinearity tailored specifically for matrix estimation. Our algorithm typically converges exponentially fast, offering a significant speedup over previously suggested matrix recovery algorithms, such as iterative solvers for nuclear norm minimization (NNM). It is well known that there is a recovery tradeoff between the information content of the objectX0to be recovered (specifically, its matrix rank r) and the number of linear measurements n from which recovery is to be attempted. The precise tradeoff between r and n, beyond which recovery by a given algorithm becomes possible, traces the so-called phase transition curve of that algorithm in the(r,n)plane. The phase transition curve of our algorithm is noticeably better than that of NNM. Interestingly, it is close to the information-theoretic lower bound for the minimal number of measurements needed for matrix recovery, making it not only state of the art in terms of convergence rate, but also near optimal in terms of the matrices it successfully recovers.

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Li, Chi-Kwong, and Yiu-Tung Poon. "Diagonals and Partial Diagonals of Sum of Matrices." Canadian Journal of Mathematics 54, no. 3 (June 1, 2002): 571–94. http://dx.doi.org/10.4153/cjm-2002-020-1.

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AbstractGiven a matrix A, let (A) denote the orbit of A under a certain group action such as(1)U(m) ⊗ U(n) acting on m × n complex matrices A by (U, V) * A = UAVt,(2)O(m) ⊗ O(n) or SO(m) ⊗ SO(n) acting on m × n real matrices A by (U, V) * A = UAVt,(3)U(n) acting on n × n complex symmetric or skew-symmetric matrices A by U * A = UAUt,(4)O(n) or SO(n) acting on n × n real symmetric or skew-symmetric matrices A by U * A = UAUt.Denote bythe joint orbit of the matrices A1, … , Ak. We study the set of diagonals or partial diagonals of matrices in (A1, … , Ak), i.e., the set of vectors (d1, … , dr) whose entries lie in the (1, j1), … , (r, jr) positions of a matrix in (A1, … , Ak) for some distinct column indices j1, … , jr . In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of A1, … , Ak. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit (A1, … , Ak) of complex Hermitian matrices under the action of unitary similarities are also discussed.

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WANG, KE. "RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES UP TO THE EDGE." Random Matrices: Theory and Applications 01, no. 01 (January 2012): 1150005. http://dx.doi.org/10.1142/s2010326311500055.

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We study the universality of the eigenvalue statistics of the covariance matrices [Formula: see text] where M is a large p × n matrix with independent entries that have mean zero, variance one and sufficiently high finite moments. In particular, as an application, we prove a variant of universality results regarding the smallest singular value of Mp, n. This paper is an extension of the results of Tao and Vu (2009) from the bulk of the spectrum up to the edge.

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