Showing papers on "Matrix analysis published in 2016"
TL;DR: The main purpose of the addressed problem is to design a non-fragile state estimator for the recurrent delayed neural networks such that the dynamics of the estimation error converges to the equilibrium asymptotically irrespective of the admissible parameter variations with the estimator gains.
93 citations
TL;DR: In this paper, it was shown that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its norm.
Abstract: We prove that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its $${\ell_2}$$
norm. Our results pertain to a wide class of random matrices, including matrices with independent entries, symmetric and skew-symmetric matrices, as well as some other naturally arising ensembles. The matrices can be real and complex; in the latter case we assume that the real and imaginary parts of the entries are independent.
60 citations
29 Feb 2016
TL;DR: This paper provides a compiler that translates a given basic linear algebra computation on structured matrices into optimized C code, optionally vectorized with intrinsics, and is extensible to a much larger set including blocked structures.
Abstract: Many problems in science and engineering are in practice modeled and solved through matrix computations. Often, the matrices involved have structure such as symmetric or triangular, which reduces the operations count needed to perform the computation. For example, dense linear systems of equations are solved by first converting to triangular form and optimization problems may yield matrices with any kind of structure. The well-known BLAS (basic linear algebra subroutine) interface provides a small set of structured matrix computations, chosen to serve a certain set of higher level functions (LAPACK). However, if a user encounters a computation or structure that is not supported, she loses the benefits of the structure and chooses a generic library. In this paper, we address this problem by providing a compiler that translates a given basic linear algebra computation on structured matrices into optimized C code, optionally vectorized with intrinsics. Our work combines prior work on the Spiral-like LGen compiler with techniques from polyhedral compilation to mathematically capture matrix structures. In the paper we consider upper/lower triangular and symmetric matrices but the approach is extensible to a much larger set including blocked structures. We run experiments on a modern Intel platform against the Intel MKL library and a baseline implementation showing competitive performance results for both BLAS and non-BLAS functionalities.
43 citations
TL;DR: In this paper, error bounds for the linear complementarity problem when the involved matrix is a B-Nekrasov matrix are presented, and numerical examples show the sharpness and applicability of the bounds.
Abstract: The class of B-Nekrasov matrices is a subclass of P-matrices that contains Nekrasov Z-matrices with positive diagonal entries as well as B-matrices. Error bounds for the linear complementarity problem when the involved matrix is a B-Nekrasov matrix are presented. Numerical examples show the sharpness and applicability of the bounds.
35 citations
Dissertation•
30 Nov 2016
TL;DR: Algorithms for a problem of finding relations in one or several variables generalizes that of computing a solution to a system of linear modular equations over a polynomial ring, including in particular the computation of Hermite-Pade approximants and bivariate interpolants.
Abstract: In this thesis, we study algorithms for a problem of finding relations in one or several
variables. It generalizes that of computing a solution to a system of linear modular
equations over a polynomial ring, including in particular the computation of Hermite-Pade
approximants and bivariate interpolants. Rather than a single solution, we aim at
computing generators of the solution set which have good properties.
Precisely, the input of our problem consists of a finite-dimensional module given by
the action of the variables on its elements, and of some elements of this module; the goal
is to compute a Grobner basis of the module of syzygies between these elements. In terms
of linear algebra, the input describes a matrix with a type of Krylov structure, and the
goal is to compute a compact representation of a basis of the nullspace of this matrix.
We propose several algorithms in accordance with the structure of the multiplication
matrices which specify the action of the variables. In the case of a Jordan matrix, we
accelerate the computation of multivariate interpolants under degree constraints; our
result for a Frobenius matrix leads to a faster algorithm for computing normal forms of
univariate polynomial matrices. In the case of several dense matrices, we accelerate the
change of monomial order for Grobner bases of multivariate zero-dimensional ideals.
29 citations
Posted Content•
TL;DR: In this article, the authors derived new relations for the approximate solution, the residual and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure.
Abstract: In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton-Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure. The new results provide theoretical ground for recently proposed modifications of projection methods onto rational Krylov subspaces.
28 citations
TL;DR: In this paper, a generalized statistical linearization technique is developed for determining approximately the stochastic response of nonlinear dynamic systems with singular matrices, and a family of optimal and response-dependent equivalent linear matrices is derived.
Abstract: A generalized statistical linearization technique is developed for determining approximately the stochastic response of nonlinear dynamic systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Specifically, relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, a family of optimal and response-dependent equivalent linear matrices is derived. This set of equations in conjunction with a generalized excitation-response relationship for linear systems leads to an iterative determination of the system response mean vector and covariance matrix. Further, it is proved ...
28 citations
TL;DR: The well-known classes of EP matrices and normal matrices are defined by the matrices that commute with their Moore-Penrose inverse and with their conjugate transpose, respectively.
Abstract: The well-known classes of EP matrices and normal matrices are defined by the matrices that commute with their Moore–Penrose inverse and with their conjugate transpose, respectively. This paper investigates the class of m-EP matrices and m-normal matrices that provide a generalization of EP matrices and normal matrices, respectively, and analyses both of them for their properties and characterizations.
27 citations
TL;DR: It is shown that a subclass of almost fully copositive matrices intorduced in (Linear Algebra Appl 400:243–252 2005) with $$Q_{0}$$Q0-property is captured by sufficient matrices introduced by Cottle et al.
Abstract: In this article, we study the properties of some matrix classes using principal pivot transform (PPT). These matrices with some additional conditions have nonnegative principal minors. We show that a subclass of almost fully copositive matrices intorduced in (Linear Algebra Appl 400:243–252 2005) with $$Q_{0}$$
-property is captured by sufficient matrices introduced by Cottle et al. in (Linear Algebra Appl 114/115:231–249 1989) and the solution set of a linear complementarity problem is the same as the set of Karush–Kuhn–Tucker stationary points of the corresponding quadratic programming problem. We introduce some more PPT based matrix classes in continuation of (Linear Algebra Appl 400:243–252 2005) and study the properties of these classes.
27 citations
TL;DR: It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily.
Abstract: If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large (81×81) but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar-Stokes parameters.
25 citations
TL;DR: In this article, a normalized characteristic matrix and the influence of the property of random coded patterns on GI was investigated based on the theory of matrix analysis and the reconstruction feature of fluctuation correlation ghost imaging (GI).
Abstract: According to the reconstruction feature of fluctuation-correlation ghost imaging (GI), we define a normalized characteristic matrix and the influence of the property of random coded patterns on GI is investigated based on the theory of matrix analysis Both simulative and experimental results demonstrate that for different random coded patterns, the quality of fluctuation-correlation GI can be predicted by some parameters extracted from the normalized characteristic matrix, which suggests its potential application in the optimization of random coded patterns for GI system
Journal Article•
TL;DR: In this article, the authors consider intervals of real matrices with respect to partial orders and the problem of inferring from some exposed matrices lying on the boundary of such an interval that all the matrices taken from the interval possess a certain property.
Abstract: This paper considers intervals of real matrices with respect to partial orders and the problem to infer from some exposed matrices lying on the boundary of such an interval that all real matrices taken from the interval possess a certain property. In many cases such a property requires that the chosen matrices have an identically signed inverse. We also briefly survey related problems, e.g., the invariance of matrix properties under entry-wise perturbations.
TL;DR: In this paper, some properties of Toeplitz matrices with respect to different matrix products are investigated and some results regarding circulant matrices, skew-circulantMatrices and approximation by Toe Platz matrices over the field of complex numbers are given.
Abstract: In this paper, we investigate some properties of Toeplitz matrices with respect to different matrix products. We also give some results regarding circulant matrices, skew-circulant matrices and approximation by Toeplitz matrices over the field of complex numbers.
TL;DR: A theoretical study extending L-structured real matrices to L- Structured quaternion matrices is presented, and their applications in quaternions matrix equations are discussed.
Abstract: This paper focuses on L-structured quaternion matrices. L-structured real matrices, conditions for the existence of solutions and the general solution of linear matrix equations were studied in the paper [Magnus JR. L-structured matrices and linear matrix equations, Linear Multilinear Algebra 1983;14:67–88]. In this paper, we present a theoretical study extending L-structured real matrices to L-structured quaternion matrices, and introduce some L-structured quaternion matrices. Based on them, we then discuss their applications in quaternion matrix equations.
Posted Content•
TL;DR: In this paper, a semi-tensor product (STP) of matrices is introduced, which can be seen as an extension of the classical matrix product to two arbitrary matrices.
Abstract: A new matrix product, called the semi-tensor product (STP), is briefly reviewed. The STP extends the classical matrix product to two arbitrary matrices. Under STP the set of matrices becomes a monoid (semi-group with identity). Some related structures and properties are investigated. Then the generalized matrix addition is also introduced, which extends the classical matrix addition to a class of two matrices with different dimensions.
Motivated by STP of matrices, two kinds of equivalences of matrices (including vectors) are introduced, which are called matrix equivalence (M-equivalence) and vector equivalence (V-equivalence) respectively. The lattice structure has been established for each equivalence. Under each equivalence, the corresponding quotient space becomes a vector space. Under M-equivalence, many algebraic, geometric, and analytic structures have been posed to the quotient space, which include (i) lattice structure; (ii) inner product and norm (distance); (iii) topology; (iv) a fiber bundle structure, called the discrete bundle; (v) bundled differential manifold; (vi) bundled Lie group and Lie algebra. Under V-equivalence, vectors of different dimensions form a vector space ${\cal V}$, and a matrix $A$ of arbitrary dimension is considered as an operator (linear mapping) on ${\cal V}$. When $A$ is a bounded operator (not necessarily square but includes square matrices as a special case), the generalized characteristic function, eigenvalue and eigenvector etc. are defined.
In one word, this new matrix theory overcomes the dimensional barrier in certain sense. It provides much more freedom for using matrix approach to practical problems.
TL;DR: In this paper, an eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with regard to a skew-symmetric form.
Abstract: An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behaviour of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients.
TL;DR: It is found that there is no great advantage in using quantum mechanics ordering rather than the optical ordering of the Stokes parameters in matrix manipulation, as has been claimed elsewhere.
Abstract: It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system. We introduce forms for the matrices for transforming between the column vector forms of these different matrices. A review of matrix algebra is presented. We find that there is no great advantage, from the point of view of matrix manipulation, in using quantum mechanics ordering rather than the optical ordering of the Stokes parameters, as has been claimed elsewhere.
TL;DR: In this paper, it was shown that a map is additive if and only if for all, the set of all invertible matrices over a field is the ring of all matrices.
Abstract: Let be the ring of all matrices over a field . It is proved that a map is additive if and only if for all , the set of all invertible matrices over .
TL;DR: There is a complexity dichotomy for Distinct Vectors based on the maximum (H) and the minimum (h) pairwise Hamming distance between matrix rows, which can be solved in polynomial time for binary matrices.
TL;DR: Having all matrix operations available, a much larger class of problems can be treated than by the restriction to matrix-vector multiplications, and the LU decomposition can be used to construct fast iterations for solving linear systems.
Abstract: Usually, one avoids numerical algorithms involving operations with large, fully populated matrices. Instead, one tries to reduce all algorithms to matrix-vector multiplications involving only sparse matrices. The reason is the large number of floating point operations; e.g., $\mathcal {O}(n^{3})$
for the multiplication of two general n × n matrices. The hierarchical matrix (
$\mathcal {H}$
-matrix) technique provides tools to perform the matrix operations approximately in almost linear work $\mathcal {O}(n\log ^{\ast }n)$
. The approximation errors are nevertheless acceptable, since large-scale matrices are usually obtained from discretisations which anyway contain a discretisation error. Adjusting the approximation error to the discretisation error yields the factor $\mathcal {O}(\log ^{\ast }n).$
The operations enabled by the $\mathcal {H}$
-matrix technique are not only the matrix addition and multiplication but also the matrix inversion and the LU or Cholesky decomposition. The positive statements from above do not hold for all matrices, but they are valid for the important class of matrices originating from standard discretisations of elliptic partial differential equations or the related integral equations. An important aspect is the fact that the algorithms can be applied in a black-box fashion. Having all matrix operations available, a much larger class of problems can be treated than by the restriction to matrix-vector multiplications. The LU decomposition can be used to construct fast iterations for solving linear systems. Also eigenvalue problems can be treated. The computation of matrix-valued functions is possible (e.g., the matrix exponential function) as well as the solution of matrix equations (e.g., of the Riccati equation).
01 Jan 2016
TL;DR: In this article, the authors introduce matrices, their representation using lists and to some of the matrix operations which Mathematica is able to do, including matrix multiplication, matrix transposition and matrix multiplication.
Abstract: Last week we did vector operations with lists. This week we introduce you to matrices, their representation using lists and to some of the matrix operations which Mathematica is able to do. Let us first see how to represent matrices in Mathematica as a list. Type " m={{a,b}, {c,d}} ". Now " type MatrixForm[m] ". Note that all of the Math-ematica operations must be applied to the list form of the matrix (not the matrix form). You can see that you get the matrix m with its elements a, b, c, d in the usual form. You can think of this matrix as consisting of two row vectors (a,b) and (c,d). Type " m[[1]] " and check you get the first row vector (a,b). Now type " m[[1,1]] " ; this will give you the 1st element of the first vector, namely a (notice that " m[[1]][[1]] " also does the same thing). Likewise, to access the element d, type " m[[2,2]] ". As you have done with vectors, you can perform algebraic operations on matrices. You can multiply a matrix with a vector. To see this, type " r={x,y} ". In order to take a dot product of the matrix m with this vector r, Type " m.r " (or Dot[m,r]). Now type " Dimensions[m] ". The output (2,2) verifies that the matrix m is a 2 × 2 matrix. At times, you need to get the transpose of a matrix, which is obtained by exchanging its off-diagonal elements (in this case the elements c and d). Type " t=Transpose[m] ". Now type " MatrixForm[t] ". You see that the matrix has diagonal elements the same but the elements c and d got interchanged with respect to the original matrix m. Often we require the determinant of a matrix, which is a scalar quantity constructed from the elements. Type " Det[m] " which will give you the determinant of the matrix m. Now type " Det[t] " and verify that the determinant is the same for the transposed matrix. A diagonal matrix has all off-diagonal elements set to zero. Type " Di-agonalMatrix[{e,f}] ". Now type " MatrixForm[DiagonalMatrix[{e,f}]] " (you can type this by taking the cursor to the end of the output and pushing the return key. Mathematica will immediately give you a replica of the output 1
Abstract: Let be a non-negative matrix. Denote by , the infimum of those satisfying the following inequality:where , and is the Fibonacci numbers sequence and is a decreasing, non-negative sequence of real numbers. In this paper, first, the Fibonacci weighted sequence space is introduced. Then, we focus on the evaluation of , where is the Hausdorff matrix operator or the Norlund matrix operator or the transpose of the Norlund matrix operator. For the case of Hausdorff matrices, a Hardy-type formula is established as an upper estimate. Also, a general upper estimate is established for the case of Norlund matrices and their transpose. In particular, we apply our results to Cesaro matrices, Holder matrices, Euler matrices and Gamma matrices.
Posted Content•
TL;DR: In this paper, a lower bound on the rank of design matrices with block (or matrix) entries has been established, extending the bounds proved in {BDWY12,DSW12} for the scalar case.
Abstract: Design matrices are sparse matrices in which the supports of different columns intersect in a few positions Such matrices come up naturally when studying problems involving point sets with many collinear triples In this work we consider design matrices with block (or matrix) entries Our main result is a lower bound on the rank of such matrices, extending the bounds proved in {BDWY12,DSW12} for the scalar case As a result we obtain several applications in combinatorial geometry The first application involves extending the notion of structural rigidity (or graph rigidity) to the setting where we wish to bound the number of `degrees of freedom' in perturbing a set of points under collinearity constraints (keeping some family of triples collinear) Other applications are an asymptotically tight Sylvester-Gallai type result for arrangements of subspaces (improving {DH16}) and a new incidence bound for high dimensional line/curve arrangements
The main technical tool in the proof of the rank bound is an extension of the technique of matrix scaling to the setting of block matrices We generalize the definition of doubly stochastic matrices to matrices with block entries and derive sufficient conditions for a doubly stochastic scaling to exist
TL;DR: Eigenvalue-eigenvector structure of Schoenmakers–Coffey matrices via Toeplitz technology and applications and applications.
01 Jan 2016
TL;DR: General relations between the terms of the sequences generated by linear homogeneous recurrence equations, for which the corresponding number sequences have some interesting number-theoretic properties are determined.
Abstract: Applying the apparatus of triangular matrices, this paper determines general relations between the terms of the sequences generated by linear homogeneous recurrence equations. We single out, in particular, the class of normal linear recurrence equations, for which the corresponding number sequences have some interesting number-theoretic properties.
TL;DR: In this article, the Cauchon algorithm is applied to nonsingular totally non-positive matrices and a new determinantal test for these matrices is derived.
Abstract: We consider classes of -by- sign regular matrices, i.e. of matrices with the property that all their minors of fixed order have one specified sign or are allowed also to vanish, . If the sign is nonpositive for all , such a matrix is called totally nonpositive. The application of the Cauchon algorithm to nonsingular totally nonpositive matrices is investigated and a new determinantal test for these matrices is derived. Also matrix intervals with respect to the checkerboard ordering are considered. This order is obtained from the usual entry-wise ordering on the set of the -by- matrices by reversing the inequality sign for each entry in a checkerboard fashion. For some classes of sign regular matrices, it is shown that if the two bound matrices of such a matrix interval are both in the same class then all matrices lying between these two bound matrices are in the same class, too.
01 Nov 2016
TL;DR: Experimental results show that the proposed 3-D motion recovery method recovers the corrupted skeleton joints, achieving accurate and smooth reconstructions even for complicated motions.
Abstract: Skeleton tracking is a useful and popular application of Kinect. However, it cannot provide accurate reconstructions for complex motions, especially in the presence of occlusion. This paper proposes a new 3-D motion recovery method based on low-rank matrix analysis to correct invalid or corrupted motions. We address this problem by representing a motion sequence as a matrix, and introducing a convex low-rank matrix recovery model, which fixes erroneous entries and finds the correct low-rank matrix by minimizing nuclear norm and norm of constituent clean motion and error matrices. Experimental results show that our method recovers the corrupted skeleton joints, achieving accurate and smooth reconstructions even for complicated motions.
TL;DR: In this paper, the relation between product decomposition of singular matrices into products of idempotent matrices and products of invertible matrices was studied. But the relation was not investigated.
Abstract: We study the relations between product decomposition of singular matrices into products of idempotent matrices and product decomposition of invertible matrices into elementary ones.