Showing papers on "Matrix analysis published in 2004"
TL;DR: This paper presents a new technique for the passivity enforcement of linear time-invariant multiport systems in state-space form based on a study of the spectral properties of related Hamiltonian matrices, aimed at the displacement of the imaginary eigenvalues of the Hamiltonian matrix.
Abstract: This paper presents a new technique for the passivity enforcement of linear time-invariant multiport systems in state-space form. This technique is based on a study of the spectral properties of related Hamiltonian matrices. The formulation is applicable in case the system input-output transfer function is in admittance, impedance, hybrid, or scattering form. A standard test for passivity is first performed by checking the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues the system is not passive. In such a case, a new result based on first-order perturbation theory is presented for the precise characterization of the frequency bands where passivity violations occur. This characterization is then used for the design of an iterative perturbation scheme of the state matrices, aimed at the displacement of the imaginary eigenvalues of the Hamiltonian matrix. The result is an effective algorithm leading to the compensation of the passivity violations. This procedure is very efficient when the passivity violations are small, so that first-order perturbation is applicable. Several examples illustrate and validate the procedure.
377 citations
Dissertation•
01 Jan 2004
TL;DR: In this paper, Behbahani et al. showed that the OD(l2; 1,1, 1, 9) is the only orthogonal design constructible from 16 circulant matrices, whenever n > 1 is an odd integer, and showed that for each integer n for which 4n is the order of a Hadamard matrix and 8n 2 + 1 is a prime, there is a productive regular matrix of order 16n 2 (8n 2 ) 2.
Abstract: On Orthogonal Matrices Majid Behbahani Depar tment of Mathemat i c s and Computer Science Universi ty of Lethbridge M. Sc. Thesis , 2004 Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. The most desirable class of matrices for our purpose are orthogonal designs constructible from 16 circulant matrices. In studying these ma trices, we show that the OD(l2; 1,1,1, 9) is the only orthogonal design constructible from 16 circulant matrices of type OD(4n; 1,1, l , 4n — 3), whenever n > 1 is an odd integer. We then use an exhaustive search to show that the only orthogonal design con structible from 16 circulant matrices of order 12 on 4 variables is the OD(12; 1 ,1,1, 9). It is known that by using of T-matrices and orthogonal designs constructible from 16 circulant matrices one can produce an infinite family of orthogonal designs. To com plement our studies we reproduce an important recent construction of T-matrices by Xia and Xia. We then turn our attention to the applications of orthogonal matrices. In some recent works productive regular Hadamard matrices are used to construct many new infinite families of symmetric designs. We show that for each integer n for which 4n is the order of a Hadamard matrix and 8n 2 — 1 is a prime, there is a productive regular Hadamard matrix of order 16n 2 (8n 2 — l ) 2 . As a corollary, we get many new infinite classes of symmetric designs whenever either of 4n(8n 2 — 1) — 1, 4n(8n 2 — 1) + 1 is a prime power. We also review some other constructions of productive regular Hadamard matrices which are related to our work.
138 citations
Posted Content•
TL;DR: In this paper, the authors developed an abstract look at linear optical networks from the viewpoint of combinatorics and permanents, and they showed that the calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents.
Abstract: We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents. An implication of this remarkable fact is that all calculations that are based on evaluating matrix elements are generically computationally hard. Moreover, quantum mechanics provides simpler derivations of certain matrix analysis results which we exemplify by showing that the permanent of any unitary matrix takes its values across the unit disk in the complex plane.
116 citations
01 Jan 2004
TL;DR: It turns out that, by employing the generalized S-procedure, one can derive smaller size of LMIs so that the computational burden can be reduced.
Abstract: In this note, we propose necessary and sufficient conditions for the asymptotic stability analysis of two-dimensional (2-D) systems in terms linear matrix inequalities (LMIs). By introducing a guardian map for the set of Schur stable complex matrices, we first reduce the stability analysis problems into nonsingularity analysis problems of parameter-dependent complex matrices. Then, by means of the discrete-time positive real lemma and the generalized S-procedure, we derive LMI-based conditions that enable us to analyze the asymptotic stability in an exact (i.e., nonconservative) fashion. It turns out that, by employing the generalized S-procedure, we can derive smaller size of LMIs so that the computational burden can be reduced
83 citations
TL;DR: This paper is a continuation of the 2004 paper “Max-algebra and pairwise comparison matrices”, in which the max-eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix.
75 citations
TL;DR: In this paper, the authors consider n-by-n matrices with a strictly dominant positive eigenvalue of multiplicity 1 and associated positive left and right eigenvectors.
Abstract: We consider those n-by-n matrices with a strictly dominant positive eigenvalue of multiplicity 1 and associated positive left and right eigenvectors. Such matrices may have negative entries and generalize the primitive matrices in important ways. Several ways of constructing such matrices, including a very geometric one, are discussed. This paper grew out of a recent survey talk about nonnegative matrices by the first author and a joint paper, with others, by the second author about the symmetric case [Tarazaga et al. (2001) Linear Algebra Appl. 328: 57].
68 citations
TL;DR: In this paper, the influence of the preparation of an open quantum system on its reduced time evolution was studied and it was shown that the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state.
Abstract: We study the influence of the preparation of an open quantum system on its reduced time evolution. In contrast to the frequently considered case of an initial preparation where the total density matrix factorizes into a product of a system density matrix and a bath density matrix the time evolution generally is no longer governed by a linear map nor is this map affine. Put differently, the evolution is truly nonlinear and cannot be cast into the form of a linear map plus a term that is independent of the initial density matrix of the open quantum system. As a consequence, the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state. The general results are elucidated with the example of two interacting spins prepared at thermal equilibrium with one spin subjected to an external field. The second spin represents the environment. The field allows the preparation of mixed density matrices of the first spin that can be represented as a convex combination of two limiting pure states, i.e., the preparable reduced density matrices make up a convex set. Moreover, the map from these reduced density matrices onto the corresponding density matrices of the total system is affine only for vanishing coupling between the spins. In general, the set of the accessible total density matrices is nonconvex.
66 citations
01 Jan 2004
TL;DR: In this paper, it was shown that a linear matrix corresponds to a compression space if and only if its rank over both rational function fields and free field (noncommutative) is equal.
Abstract: A space of matrix of low rank is a vector space of rectangular matrices whose maximum rank is stricly smaller than the number of rows and the numbers of columns Among these are the compression spaces, where the rank condition is garanteed by a rectangular hole of 0's of appropriate size Spaces of matrices are naturally encoded by linear matrices The latter have a double existence: over the rational function field, and over the free field (noncommutative) We show that a linear matrix corresponds to a compression space if and only if its rank over both fields is equal We give a simple linear-algebraic algorithm in order to decide if a given space of matrices is a compression space We give inequalities relating the commutative rank and the noncommutative rank of a linear matrix
46 citations
TL;DR: In this article, the authors studied a class of holomorphic matrix models where the integrals are taken over middle-dimensional cycles in the space of complex square matrices and the distribution of eigenvalues is given by a measure with support on a collection of arcs in the complex planes.
39 citations
TL;DR: In this paper, the authors classify families of square matrices up to the following natural equivalence, and obtain a list of all the corresponding simple mappings (that is, those that do not involve adjacent moduli).
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.
38 citations
TL;DR: In this paper, the lower and upper triangular factors of the inverse of the Vandermonde matrix are deduced using symmetric functions and combinatorial identities, which leads to q-binomial and q-Stirling matrices.
TL;DR: It is shown that the advantageous properties of totally unimodular matrices with respect to integral polyhedra can be carried over to rational k-regular matrices, and it is proved that a matrix A is k- regular if and only if the polyhedron P(A,b)={x:x?0,Ax≤b} is integral for all integral vectors b.
TL;DR: In this paper, an orthogonal-polynomials approach for random matrices with orthogonality or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, was developed, where the representation of the correlation functions in these matrix models makes use of matrix kernels.
Abstract: In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.
TL;DR: In this paper, the authors explored the mathematical properties of extreme eigenfunctions of geographic weights matrices used in spatial statistics, and applications of these properties are presented, and three theorems pertain to the popular binary geographic weights matrix based upon a planar graph.
TL;DR: In this article, it was shown that the necessary and sufficient conditions on the Jordan form of a seminonnegative matrix, identified by Zaslavsky and McDonald, are in fact necessary and necessary conditions on every eventually nonnegative matrix.
TL;DR: In this paper, an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices is derived for the case of positive definite matrix.
Abstract: We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.
27 Jun 2004
TL;DR: Ye et al. as mentioned in this paper compared principal component analysis (PCA) and subspace linear discriminant analysis (LDA) for face recognition under unconstrained illuminations (FR/I) and provided a relatively comprehensive view on the performance of linear projection methods in FR/I problems.
Abstract: Face recognition under unconstrained illuminations (FR/I) received extensive study because of the existence of illumination subspace. P. Belhumer et al. (1996) presented a study on the comparison between principal component analysis (PCA) and subspace linear discriminant analysis (LDA) for this problem. PCA and subspace LDA are two well-known linear projection methods that can be characterized as trace optimization on scatter matrices. Generally, a linear projection method can be derived by applying a specific matrix analysis technique on specific scatter matrices under some optimization criterion. Several novel linear projection methods were proposed recently using generalized singular value decomposition or QR decomposition matrix analysis techniques [H. Park, et al., 2003], [J. Ye and Q. Li, 2004]. In this paper, we present a comparative study on these linear projection methods in FR/I. We further involve multiresolution analysis in the study. Our comparative study is expected to give a relatively comprehensive view on the performance of linear projection methods in FR/I problems.
01 May 2004
TL;DR: In this article, a vector spherical-wave source scattering-matrix description of electromagnetic antennas is formulated, and reciprocity and power conservation are used to derive relations between the antenna reflection, receiving and scattering coefficients that constitute the scattering matrix.
Abstract: : The scattering-matrix analysis of linear periodic arrays of electric dipoles given in this report was suggested by the corresponding analysis performed by Yaghjian for linear periodic arrays of electro-acoustic monopole transducers. The extension to electromagnetic antennas reported here requires the use of the more complicated vector spherical wave functions instead of scalar spherical wave functions. However, no new concepts are needed for our treatment. A general vector spherical-wave source scattering-matrix description of electromagnetic antennas is formulated, and reciprocity and power conservation are used to derive relations between the antenna reflection, receiving and scattering coefficients that constitute the scattering matrix. When the antennas are electrically small dipoles only two vector spherical wave modes, one incoming and one outgoing, are required to describe the antennas. The scattering matrix formulation is applied to analyze infinite and finite linear periodic arrays of short electric dipoles perpendicular to the array axis. For an infinite periodic linear array of short dipoles, the scattering-matrix analysis leads to a simple implicit transcendental equation for the propagation constant beta of the traveling wave in terms of the normalized separation distance kd and the phase We of the effective scattering coefficient of the array elements. Interestingly, for certain values of We and kd there can be two different values of beta. The normalized separation distance and phase of the effective scattering coefficient also prove to be the only critical variables in the N x N matrix equation for the N radiation coefficients that is derived for a finite linear array of N electric dipoles. Resonances in the curves of total power radiated versus kd for a finite array excited with one feed element demonstrate the existence of the traveling waves predicted for the corresponding infinite array.
TL;DR: In this paper, the authors give a complete description of the invertible matrices over infinities and the necessary and sufficient conditions for an infinitude matrix to be invertable.
TL;DR: It is shown that every n × n matrix over a field of characteristic zero is a linear combination of three idempotent matrices, and it is proved that both 2×2 matrices and complex 3×3 matrices are linear combinations of two Idempotents.
TL;DR: In this paper, the authors studied the power sequence of Inclines, the additively idempotent semirings in which products are less than or equal to factors.
TL;DR: In this article, the algebra of all n×n complex matrices is studied and a surjective mapping is defined satisfying if and only if, then there exists a nonzero scalar c, an invertible matrix T, a function, and an automorphism f of the field such that either, or.
Abstract: Let n≥3 and let Mn be the algebra of all n×n complex matrices. If is a surjective mapping satisfying if and only if , then there exist a nonzero scalar c, an invertible matrix T, a function , and an automorphism f of the field , such that either , or .
TL;DR: The focus is on the analogues of singular value and CS (cos -- sin) decompositions for general H-unitary and Lorentz matrices, and on the Analogues of Jordan form, in a suitable basis with certain orthonormality properties, for diagonalizable H- unitary andlorentzMatrices.
Abstract: Many properties of H-unitary and Lorentz matrices are derived using elementary methods. Complex matrices that are unitary with respect to the indefinite inner product induced by an invertible Hermitian matrix H are called H-unitary, and real matrices that are orthogonal with respect to the indefinite inner product induced by an invertible real symmetric matrix are called Lorentz. The focus is on the analogues of singular value and CS (cos -- sin) decompositions for general H-unitary and Lorentz matrices, and on the analogues of Jordan form, in a suitable basis with certain orthonormality properties, for diagonalizable H-unitary and Lorentz matrices. Several applications are given, including connected components of Lorentz similarity orbits, products of matrices that are simultaneously positive definite and H-unitary, products of reflections, and stability and robust stability.
Book•
12 Aug 2004
TL;DR: In this paper, the authors present a comprehensive review of the application of linear algebra in the context of MATLAB for linear algebra with a focus on the following: 1. Linear Equations and Matrices.
Abstract: 1. Linear Equations and Matrices. Linear Systems. Matrices. Dot Product and Matrix Multiplication. Properties of Matrix Operations. Matrix Transformations. Solutions of Linear Systems of Equations. The Inverse of a Matrix. LU-Factorization (Optional). 2. Applications of Linear Equations and Matrices (Optional). An Introduction to Coding. Computer Graphics. Graph Theory. Electrical Circuits. Markov Chains. Linear Economic Models. Introduction to Wavelets. 3. Determinants. Definition and Properties. Cofactor Expansion and Applications. Determinants from a Computational Point of View. 4. Vectors in Rn . Vectors in the Plane. n-Vectors. Linear Transformations. 5. Applications of Vectors in R2 and R3 (Optional). Cross Products in R3. Lines and Planes. 6. Real Vector Spaces. Real Vector Spaces. Subspaces. Linear Independence. Basis and Dimension. Homogeneous Systems. The Rank of a Matrix and Applications. Coordinates and Change of Basis. Orthonormal Bases in Rn. Orthogonal Complements. 7. Applications of Real Vector Spaces (Optional). QR-Factorization. Least Square Lines. More on Coding. 8. Eigenvalues, Eigenvectors, and Diagonalization. Eigenvalues and Eigenvectors. Diagonalization and Similar Matrices. Diagonalization of Symmetric Matrices. 9. Applications of Eigenvalues and Eigenvectors (Optional). The Fibonacci Sequence. Differential Equations (Calculus Required). Dynamical Systems (Calculus Required). Quadratic Forms. Conic Sections. Quadric Surfaces. 10. Linear Transformations and Matrices. Definition and Examples. The Kernel and Range of a Linear Transformation. The Matrix of a Linear Transformation. Introduction to Fractals (Optional). Cumulative Review of Introductory Linear Algebra. 11. Linear Programming (Optional). The Linear Programming Problem Geometric Solution. The Simplex Method. Duality. The Theory of Games. 12. MATLAB for Linear Algebra. Input and Output in MATLAB. Matrix Operations in MATLAB. Matrix Powers and Some Special Matrices. Elementary Row Operations in MATLAB. Matrix Inverses in MATLAB. Vectors in MATLAB. Applications of Linear Combinations in MATLAB. Linear Transformations in MATLAB. MATLAB Command Summary. Appendix A: Complex Numbers. Complex Numbers. Complex Numbers in Linear Algebra. Appendix B: Further Directions. Inner Product Spaces (Calculus Required). Composite and Invertible Linear Transformations. Answers to Odd-Numbered Exercises and Chapter Tests. Index.
TL;DR: In this paper, the problem of determining necessary and sufficient conditions on two sets of vectors such that there exists an n × n matrix A in the given class satisfying Axj = yj (j = 1,..., k).
Abstract: Abstract. Assume that two sets of k vectors in Rn are given, namely {x1, . . . , xk} and {y1, . . . , yk}, and a class of matrices, e.g., positive definite matrices, positive matrices, strictly totally positive matrices, or P-matrices. The question considered in this paper is that of determining necessary and sufficient conditions on these sets of vectors such that there exists an n × n matrix A in the given class satisfying Axj = yj (j = 1, . . . , k).
TL;DR: A necessary and sufficient condition for a set of matrices to be of the form {A:A ∗ HA for some hidden) invertible Hermitian matrix H is given in this paper.
TL;DR: In this article, the authors describe how to construct a real antisymmetric matrix with prescribed eigenvalues in its leading principal submatrices and a real bi-antisymetric matrix, which is defined as a matrix with eigen values in its central sub-matrices.
TL;DR: In this article, a proof of the symmetry of the system matrix arising in a class of vector potential cell methods with nonsymmetric material matrices is presented, and some remarks on how to construct other schemes with symmetric system matrices are also presented.
Abstract: In this paper, a proof of the symmetry of the system matrix arising in a class of vector potential cell methods with nonsymmetric material matrices is presented. Some remarks on how to construct other schemes with symmetric system matrices are also presented. The explicit expression of the matrix entries derived in order to prove the symmetry is also used to show that this matrix is identical to the one arising in the standard edge finite-element method. Finally, some remarks on discrete regularizations of these formulations are given.
TL;DR: In this article, the authors compared the convergence rate of weighted pseudoinverse matrices in matrix power series and matrix power products to weighted pseudo-inverse matrix expansions with positive-definite weights in infinite matrix power product of two types: positive and negative exponents.
Abstract: On the basis of the Euler identity, we obtain expansions for weighted pseudoinverse matrices with positive-definite weights in infinite matrix power products of two types: with positive and negative exponents. We obtain estimates for the closeness of weighted pseudoinverse matrices and matrices obtained on the basis of a fixed number of factors of matrix power products and terms of matrix power series. We compare the rates of convergence of expansions of weighted pseudoinverse matrices in matrix power series and matrix power products to weighted pseudoinverse matrices. We consider problems of construction and comparison of iterative processes of computation of weighted pseudoinverse matrices on the basis of the obtained expansions of these matrices.
TL;DR: Two state-space algorithms for discrete-time J-spectral factorization of possibly singular, possibly non-column-reduced, and with possible zeros at the origin para-hermitian matrices are developed by assignment of matrices in optimal LQ return difference equality.