Showing papers on "Square matrix published in 2009"

Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging

Abstract: The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered and, in particular, a new measure of fractional anisotropy called Procrustes Anisotropy is discussed.

Polar codes: Characterization of exponent, bounds, and constructions

Abstract: Polar codes were recently introduced by Arikan. They achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancellation decoding strategy. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the 2 × 2 matrix of the given equation. It was shown by Arikan and Telatar that this construction achieves an error exponent of 1/2, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the length. It was already mentioned by Arikan that in principle larger matrices can be used to construct polar codes. A fundamental question then is to see whether there exist matrices with exponent exceeding 1/2. We characterize the exponent of a given square matrix and derive upper and lower bounds on achievable exponents. Using these bounds we show that there are no matrices of size less than 15 with exponents exceeding 1/2. Further, we give a general construction based on BCH codes which for large matrix sizes achieves exponents arbitrarily close to 1 and which exceeds 1/2 for size 16.

A new investigation of the extended Krylov subspace method for matrix function evaluations

Abstract: For large square matrices A and functions f , the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate the Extended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques.

The single ring theorem

Abstract: We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure on the complex plan whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem \cite{FZ}. We also consider the case where $U_n,V_n$ are independent Haar distributed on the orthogonal group.

Rectangular random matrices, related convolution

Abstract: We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all noncommutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra. Therefore, we can define a “rectangular-free convolution”, which allows to deduce the singular values of the sum of two large independent rectangular random matrices from the individual singular values. This convolution is linearized by cumulants and by an analytic integral transform, that we called the “rectangular R-transform”.

Linearizations of singular matrix polynomials and the recovery of minimal indices

Abstract: A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equivalent matrix pencil - a process known as linearization. Two vector spaces of pencils L1(P ) and L2(P ) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ )w hen P is regular. The goal of this work is to show that most of the pencils in L1(P )a ndL2(P )a re stil l linearizations when P (λ) is a singular square matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ )f rom the minimalindices and bases of l in L1(P )a ndL2(P ). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P )= L1(P ) ∩ L2(P )w il l never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases.

Multi-Fourier spectra descriptor and augmentation with spectral clustering for 3D shape retrieval

Abstract: We propose a new method of similarity search for 3D shape models, given an arbitrary 3D shape as a query. The method features the high search performance enabled in part by our unique feature vector called Multi-Fourier Spectra Descriptor (MFSD), and in part by augmenting the feature vector with spectral clustering. The MFSD is composed of four independent Fourier spectra with periphery enhancement. It allows us to faithfully capture the inherent characteristics of an arbitrary 3D shape object regardless of the dimension, orientation, and original location of the object when it is first defined. Given a 3D shape database, the augmentation with spectral clustering is done first by computing the p-minimum spanning tree of the whole data set, where p is a number usually much less than m, the size of the whole 3D shape data set. We then define the affinity matrix, which is a square matrix of size m by m, where each element of the matrix denotes the distance between two shape objects. The distance is computed in advance by traversing the p-minimum spanning tree. The eigenvalue decomposition is then applied to the affinity matrix to reduce dimensionality of the matrix, followed by grouping into k clusters. The cluster information is kept for augmenting the search performance when a query is given. With a series of benchmark data sets, we will demonstrate that our approach outperforms previously known methods for 3D shape retrieval.

Optimal Control for Polynomial Systems Using Matrix Sum of Squares Relaxations

Abstract: This note deals with a computational approach to an optimal control problem for input-affine polynomial systems based on a state-dependent linear matrix inequality (SDLMI) from the Hamilton-Jacobi inequality. The design follows a two-step procedure to obtain an upper bound on the optimal value and a state feedback law. In the first step, a direct usage of the matrix sum of squares relaxations and semidefinite programming gives a feasible solution to the SDLMI. In the second step, two kinds of polynomial annihilators decrease the conservativeness of the first design. The note also deals with a control-oriented structural reduction method to reduce the computational effort. Numerical examples illustrate the resulting design method.

Simple square smoothing regularization operators

Abstract: Tikhonov regularization of linear discrete ill-posed prob lems often is applied with a finite differ- ence regularization operator that approximates a low-order derivative. These operators generally are represented by a banded rectangular matrix with fewer rows than columns. They therefore cannot be applied in iterative meth- ods that are based on the Arnoldi process, which requires the regularization operator to be represented by a square matrix. This paper discusses two approaches to circumvent this difficulty: zero-padding the rectangular matrices to make them square and extending the rectangular matrix to a square circulant. We also describe how to com- bine these operators by weighted averaging and with orthogonal projection. Applications to Arnoldi and Lanczos bidiagonalization-based Tikhonov regularization, as well as to truncated iteration with a range-restricted minimal residual method, are presented.

Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M -wavelets

Abstract: Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2π-periodic trigonometric polynomials with symmetry satisfying \(\mathbf {p}\overline{\mathbf{p}}^{T}=1\) . Matrix extension plays a fundamental role in many areas such as electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a step-by-step simple algorithm to derive a desired square matrix P from a given row vector p of 2π-periodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor M, we shall present two families of compactly supported symmetric orthonormal complex M-wavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linear-phase moments and vanishing moments, which are desirable properties in numerical algorithms.

Non-commutative desingularization of determinantal varieties, I

Abstract: We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.

The Stable Perturbation of the Drazin Inverse of the Square Matrices

Abstract: For any $n\times n$ complex matrix $A$, let $A^D$ and $A^\pi$ be the Drazin inverse and the spectral projector of $A$, respectively, where $A^\pi=I-AA^D$. When $A$ is singular, an $n\times n$ complex matrix $B$ is said to be a stable perturbation of $A$ if $I-A^\pi-B^\pi$ is nonsingular or, equivalently, if the matrix $B$ satisfies condition (${\cal C}_s$) recently introduced by Castro-Gonzalez, Robles, and Velez-Cerrada [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 882-897]. In the perturbation analysis of the Drazin inverse, the condition of $\Vert B-A\Vert$ being small is usually implicitly assumed in the literature. In this case, the condition of $B$ being a stable perturbation of $A$ is necessary in order to ensure the continuity of the Drazin inverse. In this paper, only under the condition that $B$ is a stable perturbation of $A$, two explicit formulas for the Drazin inverse $B^D$ and the spectral projector $B^\pi$ are provided, respectively, and some upper bounds for $\Vert B^D-A^D\Vert/\Vert A^D\Vert$ and $\Vert B^\pi-A^\pi\Vert$ are derived from these formulas under certain conditions. In the case where $\Vert B-A\Vert$ is small, numerical examples are given indicating the sharpness of these norm upper bounds. Furthermore, a numerical example is provided to illustrate that a perturbation analysis of the Drazin inverse can also be conducted, even if $\Vert B-A\Vert$ is not small.

Explicit representations of the Drazin inverse of block matrix and modified matrix

Abstract: A new result on the Drazin inverse of 2 × 2 block matrix , where A and C are square matrices are presented, extended in the case when D = 0, the well-known representation for the Drazin inverse of M, given by Hartwig, Meyer and Rose in 1977, respectively. Using that new result, an explicit representation for the Drazin inverse of a modified matrix P + RS and its generalized matrix PQ + RS are also presented.

Polar Codes: Characterization of Exponent, Bounds, and Constructions

Abstract: Polar codes were recently introduced by Ar\i kan. They achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels under a low complexity successive cancellation decoding strategy. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the $2 \times 2$ matrix $\bigl[ 1 &0 1& 1 \bigr]$. It was shown by Ar\i kan and Telatar that this construction achieves an error exponent of $\frac12$, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the length. It was already mentioned by Ar\i kan that in principle larger matrices can be used to construct polar codes. A fundamental question then is to see whether there exist matrices with exponent exceeding $\frac12$. We first show that any $\ell \times \ell$ matrix none of whose column permutations is upper triangular polarizes symmetric channels. We then characterize the exponent of a given square matrix and derive upper and lower bounds on achievable exponents. Using these bounds we show that there are no matrices of size less than 15 with exponents exceeding $\frac12$. Further, we give a general construction based on BCH codes which for large $n$ achieves exponents arbitrarily close to 1 and which exceeds $\frac12$ for size 16.

Solutions of matrix NLS systems and their discretisations: A unified treatment

Abstract: Using a bidifferential graded algebra approach to integrable partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework, by specifying a representation of the bidifferential graded algebra and imposing a reduction. By application of a general result, corresponding families of exact solutions are obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The solutions are parametrised in terms of constant matrix data subject to a Sylvester equation (which previously appeared as a rank condition in the integrable systems literature). These data exhibit a certain redundancy, which we diminish to a large extent. More precisely, we first consider more general AKNS-type systems from which two different matrix NLS systems emerge via reductions. In the continuous case, the familiar Hermitian conjugation reduction leads to a continuous matrix (including vector) NLS equation, but it is well-known that this does not work as well in the discrete cases. On the other hand there is a complex conjugation reduction, which apparently has not been studied previously. It leads to square matrix NLS systems, but works in all three cases (continuous, semi- and fully-discrete). A large part of this work is devoted to an exploration of the corresponding solutions, in particular regularity and asymptotic behaviour of matrix soliton solutions.

Non-Hermitian Random Matrix Ensembles

Abstract: This is a concise review of the complex, real and quaternion real Ginibre random matrix ensembles and their elliptic deformations. Eigenvalue correlations are exactly reduced to two-point kernels and discussed in the strongly and weakly non-Hermitian limits of large matrix size.

Edge Universality for Orthogonal Ensembles of Random Matrices

Abstract: We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.

A Multiple Secret Sharing Scheme based on Matrix Projection

Abstract: In [3], Bai et al. have proposed a multiple secret sharing scheme based on matrix projection. It is an elegant scheme with several advantages such as small share size and dynamic to secret changes. However,one of its disadvantages is that the secrets are organized in a square matrix and hence the number of secrets must be a square. So there is often a necessity to stuff dummy secrets into the secret matrix if the number of secrets is not a square.We present a new scheme based on matrix projection method that can share any number of secrets and make full use of every element of the secret matrix. The proposed scheme is as secure as Bai's scheme. Besides, the proposed scheme can also take advantage of the proactive characteristic of the Matrix Projection Method to update shares periodically to improve security.Our scheme increases the potential range of the threshold. The increment of the threshold range is even more when we are using the proactive feature of the scheme. It also further reduces the share size to a constant (equal to that of a single secret). As with Bai's scheme, our scheme is partially verifiable based on the properties of the projection matrix. The paper also summarizes and classifies typical existing secret sharing schemes.

A Useful Form of Unitary Matrix Obtained from Any Sequence of Unit 2-Norm $n$-Vectors

Abstract: Charles Sheffield pointed out that the modified Gram-Schmidt (MGS) orthogonalization algorithm for the QR factorization of $B\!\in\!\R^{n\times k}$ is mathematically equivalent to the QR factorization applied to the matrix $B$ augmented with a $k\times k$ matrix of zero elements on top. This is true in theory for any method of QR factorization, but for Householder's method it is true in the presence of rounding errors as well. This knowledge has been the basis for several successful but difficult rounding error analyses of algorithms which in theory produce orthogonal vectors but significantly fail to do so because of rounding errors. Here we show that the same results can be found more directly and easily without recourse to the MGS connection. It is shown that for any sequence of $k$ unit 2-norm $n$-vectors there is a special $(n\!+\!k)$-square unitary matrix which we call a unitary augmentation of these vectors and that this matrix can be used in the analyses without appealing to the MGS connection. We describe the connection of this unitary matrix to Householder matrices. The new approach is applied to an earlier analysis to illustrate both the improvement in simplicity and advantages for future analyses. Some properties of this unitary matrix are derived. The main theorem on orthogonalization is then extended to cover the case of biorthogonalization.

On the reflexive solutions of the matrix equation AXB+CYD=E

Abstract: A matrix P 2C n◊n is called a generalized reflection matrix if P ⁄ = P and P 2 = I. An n◊n complex matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = PAP (A = iPAP). It is well-known that the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P have many special properties and widely used in engineering and sci- entific computations. In this paper, we give new necessary and sucient conditions for the existence of the reflexive (anti-reflexive) solutions to the linear matrix equation AXB + CY D = E and derive representation of the general reflexive (anti-reflexive) solutions to this matrix equation. By using the obtained results, we investigate the reflexive (anti-reflexive) solutions of some special cases of this matrix equation.

Verified Computation of Square Roots of a Matrix

Abstract: We present methods to compute verified square roots of a square matrix $A$. Given an approximation $X$ to the square root, obtained by a classical floating point algorithm, we use interval arithmetic to find an interval matrix which is guaranteed to contain the error of $X$. Our approach is based on the Krawczyk method, which we modify in two different ways in such a manner that the computational complexity for an $n\times n$ matrix is reduced to $n^3$. The methods are based on the spectral decomposition or, in the case that the eigenvector matrix is ill conditioned, on a similarity transformation to block diagonal form. Numerical experiments prove that our methods are computationally efficient and that they yield narrow enclosures provided $X$ is a good approximation. This is particularly true for symmetric matrices, since their eigenvector matrix is perfectly conditioned.