Showing papers on "Square matrix published in 2004"

A Measure of Similarity between Graph Vertices: Applications to Synonym Extraction and Web Searching

Abstract: We introduce a concept of {similarity} between vertices of directed graphs. Let GA and GB be two directed graphs with, respectively, nA and nB vertices. We define an nB \times nA similarity matrix S whose real entry sij expresses how similar vertex j (in GA) is to vertex i (in GB): we say that sij is their similarity score. The similarity matrix can be obtained as the limit of the normalized even iterates of Sk+1 = BSkAT + BTSkA, where A and B are adjacency matrices of the graphs and S0 is a matrix whose entries are all equal to 1. In the special case where GA = GB = G, the matrix S is square and the score sij is the similarity score between the vertices i and j of G. We point out that Kleinberg's "hub and authority" method to identify web-pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them. In analogy to Kleinberg, we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix. Potential applications of our similarity concept are numerous. We illustrate an application for the automatic extraction of synonyms in a monolingual dictionary.

Matrix Exponential Gradient Updates for On-line Learning and Bregman Projection

Abstract: We address the problem of learning a symmetric positive definite matrix. The central issue is to design parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann divergence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: On-line learning with a simple square loss and finding a symmetric positive definite matrix subject to symmetric linear constraints. The updates generalize the Exponentiated Gradient (EG) update and AdaBoost, respectively: the parameter is now a symmetric positive definite matrix of trace one instead of a probability vector (which in this context is a diagonal positive definite matrix with trace one). The generalized updates use matrix logarithms and exponentials to preserve positive definiteness. Most importantly, we show how the analysis of each algorithm generalizes to the non-diagonal case. We apply both new algorithms, called the Matrix Exponentiated Gradient (MEG) update and DefiniteBoost, to learn a kernel matrix from distance measurements.

A theorem of the alternatives for the equation Ax + B|x| = b

Abstract: The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ≤ D and for each b, or the equation Ax + B 0|x| = 0 has a nontrivial solution for some matrix B 0 of a very special form, |B 0| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| = λ|x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue.

Two characterizations of inverse-positive matrices: the hawkins-simon condition and the le chatelier-braun principle ∗

Abstract: Dedicated to the late Professors David Hawkins and Hukukane Nikaido Abstract. It is shown that (a weak version of) the Hawkins-Simon condition is satisfied by any real square matrix which is inverse-positive after a suitable permutation of columns or rows. One more characterization of inverse-positive matrices is given concerning the Le Chatelier-Braun principle. The proofs are all simple and elementary.

The control of linear time-periodic systems using Floquet–Lyapunov theory

Abstract: In this paper we use Floquet–Lyapunov theory to derive the Floquet factors of the state-transition matrix of a given linear time-periodic system We show how the periodicity of one of the factors can be determined a priori using a constant matrix, which we call the Yakubovich matrix, based upon the signs of the eigenvalues of the monodromy matrix We then describe a method for the numerical computation of the Floquet factors, relying upon a boundary-value problem formulation and the Yakubovich matrix Further, we show how the invertibility of the controllability Gramian and a specific form for the feedback gain matrix can be used to derive a control law for the closed-loop system The controller can be full-state or observer-based It also allows the engineer to assign all the invariants of the system; ie the full monodromy matrix Deriving the feedback matrix requires solving a matrix integral equation for the periodic Floquet factor of the new state-transition matrix of the closed-loop system This is

Detecting short directed cycles using rectangular matrix multiplication and dynamic programming

Abstract: We present several new algorithms for detecting short fixed length cycles in digraphs. The new algorithms utilize fast rectangular matrix multiplication algorithms together with a dynamic programming approach similar to the one used in the solution of the classical chain matrix product problem. The new algorithms are instantiations of a generic algorithm that we present for finding a directed Ck, i.e., a directed cycle of length k, in a digraph, for any fixed k ≥ 3. This algorithm partitions the prospective Ck's in the input digraph G = (V,E) into O(logkV) classes, according to the degrees of their vertices. For each cycle class we determine, in O(Eck log V) time, whether G contains a Ck from that class, where ck = ck(ω) is a constant that depends only on !, the exponent of square matrix multiplication. The search for cycles from a given class is guided by the solution of a small dynamic programming problem. The total running time of the obtained deterministic algorithm is therefore O(Eck logk+1V).For C3, we get c3 = 2ω/(ω + 1)

Bounds for Analytical Functions of Matrices

Abstract: If f is an analytic function bounded on a convex domain of the complex plane and A a square matrix whose spectrum is included in this domain, the function f(A) is well defined. In this paper we study bounds for ||f(A)|| uniform with respect to the functions f bounded by 1, and uniform with respect to the matrices A whose the numerical ranges are included in the domain. We show that these bounds are attained and give explicit formulae in some 2-dimensional cases.

Floquet Theory as a Computational Tool

Abstract: We describe how classical Floquet theory may be utilized, in a continuation framework, to construct an efficient Fourier spectral algorithm for approximating periodic orbits. At each continuation step, only a single square matrix, whose size equals the dimension of the phase-space, needs to be factorized; the rest of the required numerical linear algebra just consists of back-substitutions with this matrix. The eigenvalues of this key matrix are the Floquet exponents, whose crossing of the imaginary axis indicates bifurcation and change-in-stability. Hence we also describe how the new periodic orbits created at a period-doubling bifurcation point may be efficiently computed using our approach.

On Inverse Quadratic Eigenvalue Problems with Partially Prescribed Eigenstructure

Abstract: The inverse eigenvalue problem of constructing real and symmetric square matrices M, C, and K of size $n \times n$ for the quadratic pencil $Q(\lambda) = \lambda^2 M + \lambda C + K$ so that $Q(\lambda)$ has a prescribed subset of eigenvalues and eigenvectors is considered This paper consists of two parts addressing two related but different problemsThe first part deals with the inverse problem where M and K are required to be positive definite and semidefinite, respectively It is shown via construction that the inverse problem is solvable for any k, given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided $k \leq n$ The construction also allows additional optimization conditions to be built into the solution so as to better refine the approximate pencil The eigenstructure of the resulting $Q(\lambda)$ is completely analyzedThe second part deals with the inverse problem where M is a fixed positive definite matrix (and hence may be assumed to be t

Apparatus for encoding and decoding of low-density parity-check codes, and method thereof

Abstract: An LDPC code encoding apparatus includes: a code matrix generator for generating and transmitting a parity-check matrix comprising a combination of square matrices having a unique value on each row and column thereof; an encoding means encoding block LDPC codes according to the parity-check matrix received from the code matrix generator; and a codeword selector for puncturing the encoded result of the encoding means to generate an LDPC codeword. The code matrix generator divides an information word to be encoded into block matrices having a predetermined length to generate a vector information word. The encoding means encodes the block LDPC codes using the parity-check matrix divided into the block matrices and a Tanner graph divided into smaller graphs in correspondence to the parity-check matrix.

On Families of Square Matrices

Abstract: In this paper we classify families of square matrices up to the following natural equivalence. Thinking of these families as germs of smooth mappings from a manifold to the space of square matrices, we allow arbitrary smooth changes of co-ordinates in the source and pre- and post- multiply our family of matrices by (generally distinct) families of invertible matrices, all dependent on the same variables. We obtain a list of all the corresponding simple mappings (that is, those that do not involve adjacent moduli). This is a non-linear generalisation of the classical notion of linear systems of matrices. We also make a start on an understanding of the associated geometry. 2000 Mathematics Subject Classification 58K40, 58K50, 58K60, 32S25.

The Worst-Case GMRES for Normal Matrices

Abstract: We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. This bound is sharp, i.e. it is attainable by the GMRES residual norm. The question is how to evaluate or estimate the standard bound, and if it is possible to characterize the GMRES-related quantities for which this bound is attained (worst-case GMRES). In this paper we completely characterize the worst-case GMRES-related quantities in the next-to-last iteration step and evaluate the standard bound in terms of explicit polynomials involving the matrix eigenvalues. For a general iteration step, we develop a computable lower and upper bound on the standard bound. Our bounds allow us to study the worst-case GMRES residual norm as a function of the eigenvalue distribution. For hermitian matrices the lower bound is equal to the worst-case residual norm. In addition, numerical experiments show that the lower bound is generally very tight, and support our conjecture that it is to within a factor of 4/π of the actual worst-case residual norm. Since the worst-case residual norm in each step is to within a factor of the square root of the matrix size to what is considered an “average” residual norm, our results are of relevance beyond the worst case.

On mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product

Abstract: Some mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product are established. Necessary and sufficient conditions for these laws to hold are found by the matrix rank method. Some applications and extensions of these reverse-order laws to the weighted Moore-Penrose inverse are also given. 2000 Mathematics Subject Classification: 15A03, 15A09. If A and B are a pair of invertible matrices of the same size, then the product AB is nonsingular, too, and the inverse of the product AB satisfies the reverse-order law (AB) −1 = B −1 A −1 . This law can be used to find the properties of (AB) −1 ,a s well as to simplify various matrix expressions that involve the inverse of a matrix product. However, this formula cannot trivially be extended to the Moore-Penrose inverse of matrix products. For a general m ×n complex matrix A, the Moore-Penrose inverse A † of A is the unique n ×m matrix X that satisfies the following four Penrose equations:

Interleaver and de-interleaver

Abstract: An interleaver employs a generalized method of generating a mapping. The mapping is generated for interleaving bits of a data block and associated error detection/correction information. The data block is of length N, and the length of the error detection/correction information is P. An (N+P)×(N+P) square matrix is formed and divided into sub-blocks, where one portion of the matrix is associated with error detection/correction information and another portion is associated with data of the data block. New positions in the matrix are generated in a time sequence on a sub-block by sub-block basis based on a generator seed pair and an original position seed pair. The time sequence also corresponds to positions in an output interleaved block. Once the new position sequence is generated, the matrix is populated with data and error detection/correction information based on the corresponding time sequence. A de-interleaver performs the inverse mapping of the interleaver.

Exact effective Hamiltonian theory. II. Polynomial expansion of matrix functions and entangled unitary exponential operators

Abstract: Our recent exact effective Hamiltonian theory (EEHT) for exact analysis of nuclear magnetic resonance (NMR) experiments relied on a novel entanglement of unitary exponential operators via finite expansion of the logarithmic mapping function. In the present study, we introduce simple alternant quotient expressions for the coefficients of the polynomial matrix expansion of these entangled operators. These expressions facilitate an extension of our previous closed solution to the Baker–Campbell–Hausdorff problem for SU(N) systems from N⩽4 to any N, and thereby the potential application of EEHT to more complex NMR spin systems. Similarity matrix transformations of the EEHT expansion are used to develop alternant quotient expressions, which are fully general and prove useful for evaluation of any smooth matrix function. The general applicability of these expressions is demonstrated by several examples with relevance for NMR spectroscopy. The specific form of the alternant quotients is also used to demonstrate ...

Efficient dynamic equations of 3-RPS parallel mechanism through Lagrange method

Abstract: By means of an efficient method, the generalized reduced order dynamic equation for 3-RPS parallel mechanism through Lagrange method is derived. Kinematic constraints accompanying the Lagrange method for the constrained set of generalized coordinates, introduces the Lagrange multiplier into dynamical formulation. To omit the Lagrange multipliers the natural orthogonal complement matrix of kinematic constraints' matrix should be found. To reach the natural orthogonal complement matrix, the inverse of a square matrix having order equal to the rank of a kinematic constraints' matrix should be found. For a system having many kinematic constraints like 3-RPS, the rank of the aforementioned matrix will be high. In this research it is shown that for a 3-RPS parallel manipulator, a rearranging matrix derived from the kinematic constraints on constrained coordinates in a special way will simplify the inverse calculation. Instead of inversion of a high order matrix, only inversion of some very low order matrices should be evaluated. Therefore the natural orthogonal complement matrix can be reached without the need for inversion of a high order matrix and Lagrange multipliers can be omitted again very easily.

Higher order matrix SUSY transformations in two-dimensional quantum mechanics

Abstract: The iteration procedure of supersymmetric transformations for the two-dimensional Schrodinger operator is implemented by means of the matrix form of factorization in terms of matrix 2 × 2 supercharges. Two different types of iterations are investigated in detail. The particular case of diagonal initial Hamiltonian is considered, and the existence of solutions is demonstrated. Explicit examples illustrate the construction.

The Judgement and Parallel Algorithm for Inverse M-matrixes

Abstract: A simple and convenient judging method will be given. It is used to judge whether an n×n nonnegative real matrix A is an inverse M-matrix or not. By using the method, we reduce the order of an n×n matrix A gradually until the reduced matrix can be judged by the definition of inverse M-matrixes. At the same time, aglorithm and its operation problem are also been studied.