Solution Of Linear Matrix Equation Research Articles

Iterative optimal solutions of linear matrix equations for hyperspectral and multispectral image fusing

For a linear matrix function f in X in {mathbb {R}}^{mtimes n} we consider inhomogeneous linear matrix equations f(X) = E for E ne 0 that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos’ methods in combination with the More–Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are performed that illustrate universality and efficiency of our method for dense and small data matrices, as well as for sparse and certain structured input matrices. Specifically we show how to adapt our universal method for sparse inputs and for structured data such as encountered when fusing image data sets via a Sylvester equation algorithm to obtain an image of higher resolution.

Guaranteed root mean square estimates of linear matrix equations solutions under conditions of uncertainty

The article focuses on the linear estimation problems of unknown rectangular matrices, which are solutions of linear matrix equations with the right-hand sides belonging to bounded sets. The random errors of the observation vector have zero mathematical expectation, and the correlation matrix is unknown and belongs to one of two bounded sets. Explicit expressions of the guaranteed root mean square errors of estimates for linear operators acting from the space of rectangular matrices into some vector space are given. Guaranteed quasi-minimax root mean square errors of linear estimates are obtained. As the test examples, two options for solving the problem are considered, taking into account small perturbations of known observation matrices.

Optimal Stabilization of Limkarized Systems VIA Receding Horizon Technique.(Dept.E)

The paper investigates the role of horizon length M in tuning the dynamical behaviour of discrete time systems. The paper introduces , as well as a new method for solving the reading horizon matrix differential equation, which avoids solution of linear matrix equation usually encountered. An efficient fast and rigid algorithm is developed which is well suited for on-line control using micro-processors.

Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations

Galerkin and Petrov–Galerkin methods are some of the most successful solution procedures in numerical analysis. Their popularity is mainly due to the optimality properties of their approximate solution. We show that these features carry over to the (Petrov-) Galerkin methods applied for the solution of linear matrix equations. Some novel considerations about the use of Galerkin and Petrov–Galerkin schemes in the numerical treatment of general linear matrix equations are expounded and the use of constrained minimization techniques in the Petrov–Galerkin framework is proposed.

Computing symmetric solutions of general Sylvester matrix equations via Lanczos version of biconjugate residual algorithm

Many problems in control theory can be studied by obtaining the symmetric solution of linear matrix equations. In this investigation, we deal with the symmetric solutions X, Y and Z of the general Sylvester matrix equations A1XB1+C1YD1+E1ZF1=G1,A2XB2+C2YD2+E2ZF2=G2,⋮⋮⋮⋮AtXBt+CtYDt+EtZFt=Gt.The Lanczos version of biconjugate residual (BCR) algorithm is generalized to compute the symmetric solutions of the general Sylvester matrix equations. The convergence properties of this algorithm are discussed and it is proven that it smoothly converges to the symmetric solutions of the general Sylvester matrix equations in a finite number of iterations in the absence of round-off errors. Finally, the comparative numerical results are carried out to support that the current algorithm may be more efficient than other iterative algorithms.

Rank/inertia approaches to weighted least-squares solutions of linear matrix equations

The well-known linear matrix equation AX=B is the simplest representative of all linear matrix equations. In this paper, we study quadratic properties of weighted least-squares solutions of this matrix equation. We first establish two groups of closed-form formulas for calculating the global maximum and minimum ranks and inertias of matrices in the two quadratical matrix-valued functions Q1−XP1X′ and Q2−X′P2X subject to the restriction trace[(AX−B)′W(AX−B)]=min, where both Pi and Qi are real symmetric matrices, i=1,2,W is a positive semi-definite matrix, and X′ is the transpose of X. We then use the rank and inertia formulas to characterize quadratic properties of weighted least-squares solutions of AX=B, including necessary and sufficient conditions for weighted least-squares solutions of AX=B to satisfy the quadratic symmetric matrix equalities XP1X′=Q1 an X′P2X=Q2, respectively, and necessary and sufficient conditions for the quadratic matrix inequalities XP1X′≻Q1 (≽Q1, ≺Q1, ≼Q1) and X′P2X≻Q2 (≽Q2, ≺Q2, ≼Q2) in the Löwner partial ordering to hold, respectively. In addition, we give closed-form solutions to four Löwner partial ordering optimization problems on Q1−XP1X′ and Q2−X′P2X subject to weighted least-squares solutions of AX=B.

Quadratic properties of least-squares solutions of linear matrix equations with statistical applications

Assume that a quadratic matrix-valued function \(\psi (X) = Q - X^{\prime }PX\) is given and let \(\mathcal{S} = \left\{ X\in {\mathbb R}^{n \times m} \, | \, \mathrm{trace}[\,(AX - B)^{\prime }(AX - B)\,] = \min \right\} \) be the set of all least-squares solutions of the linear matrix equation \(AX = B\). In this paper, we first establish explicit formulas for calculating the maximum and minimum ranks and inertias of \(\psi (X)\) subject to \(X \in {\mathcal S}\), and then derive from the formulas the analytic solutions of the two optimization problems \(\psi (X) =\max \) and \(\psi (X)= \min \) subject to \(X \in \mathcal{S}\) in the Lowner partial ordering. As applications, we present a variety of results on equalities and inequalities of the ordinary least squares estimators of unknown parameter vectors in general linear models.

Symmetric solutions of the coupled generalized Sylvester matrix equations via BCR algorithm

The symmetric solutions of linear matrix equations are extensively required in mathematics and engineering problems. The purpose of this paper is on deriving the biconjugate residual (BCR) algorithm for finding the least Frobenius norm symmetric solution pair (X,Y) of the coupled generalized Sylvester matrix equations{A1XB1+C1XD1+E1YF1=G1,A2XB2+C2XD2+E2YF2=G2.The convergence analysis shows that the BCR algorithm can compute the least Frobenius norm symmetric solution pair of the coupled generalized Sylvester matrix equations within a finite number of iterations in the absence of round-off errors. Finally, we give three numerical examples to illustrate the performance of the BCR algorithm.

Induced Dimension Reduction Method for Solving Linear Matrix Equations

This paper discusses the solution of large-scale linear matrix equations using the Induced Dimension reduction method (IDR(s)). IDR(s) was originally presented to solve system of linear equations, and is based on the IDR(s) theorem. We generalize the IDR(s) theorem to solve linear problems in any finite-dimensional space. This generalization allows us to develop IDR(s) algorithms to approximate the solution of linear matrix equations. The IDR(s) method presented here has two main advantages; firstly, it does not require the computation of inverses of any matrix, and secondly, it allows incorporation of preconditioners. Additionally, we present a simple preconditioner to solve the Sylvester equation based on a fixed point iteration. Several numerical examples illustrate the performance of IDR(s) for solving linear matrix equations. We also present the software implementation.

Алгоритм оперативного определения слабых звеньев сети, приводящих к нарушению статической устойчивости

Considered a computing matrix algorithm to quickly identify weaknesses that lead to the violation of the small signal stability of the linear model of the electric power system. The algorithm is based on the method of partial solutions of linear matrix equation based on the use of zero divisors (annihilators) matrices. Features of the algorithm shown in the model of the interconnection with the Jacobi matrix containing 286 units, 531 branches and 129 generators.

Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for a class of linear matrix equations

Recently, Wang et al. (2013) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method to approximate the solution of linear matrix equation. To further improve the efficiency of the HSS method, we in this paper establish a parameterized preconditioned HSS (PPHSS) iteration method by introducing two Hermitian positive definite preconditioners and two more different iteration parameters for the HSS method. The convergence properties of the PPHSS method used for solving linear matrix equation AXB=C are strictly analyzed and the quasi-optimal values of the iteration parameters for the PPHSS method are also derived. Numerical results demonstrate the feasibility and the efficiency of the PPHSS iteration method.

L-structured quaternion matrices and quaternion linear matrix equations

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.

DOUBLE ITERATIVE ALGORITHM FOR SYMMETRIC SOLUTION OF DISCRETE TIME ALGEBRA RICCATI MATRIX EQUATION

Based on the modified conjugate gradient method in computing constrained solution of general linear matrix equation,an iterative method is proposed to compute the symmetric solution of discrete time algebra Riccati matrix equation(DTARME) in optimal control system.Firstly,DTARME is processed by matrix series.Then Newton's method is applied to find symmetric solution of DTARME,and the symmetric solution or symmetric least-square solution of the linear matrix equation is solved.Finally,the modified conjugate gradient method is applied to solve the derived linear matrix equation.Therefore,a double iterative method is established to find symmetric solution of DTARME,and the corresponding convergence conclusion is given.Finally,numerical experiments show that the double iterative algorithm is effective.

Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equationQ=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method.

Expository Research Papers

The two papers in this issue have to do with matrices and sparsity—but from different points of view. Sparsity, in the first paper, means many zero elements in the matrix, while in the second paper it refers to many zero singular values, i.e., low rank. The context of the first paper, “On the Block Triangular Form of Symmetric Matrices” by Iain Duff and Bora Uçar, is the solution of linear systems of equations $Ax = b$ whose coefficient matrix A is sparse, i.e., has many zero elements. A direct method, such as Gaussian elimination, becomes more efficient if one can first permute the rows and columns of A into block triangular form. The classical method for permuting a matrix to block triangular form is due to Dulmage and Mendelsohn and dates back to 1963. The idea is to represent the matrix A as a bipartite graph whose nodes are columns and rows of A, and whose edges correspond to nonzero elements of A, and then to determine a matching of maximum cardinality in this graph. That means determining a maximal number of nonzero elements no two of which belong to the same row or column. Duff and Uçar analyze the block triangular form for a particular class of square matrices A: These matrices are structurally symmetric, i.e., their zero/nonzero structure is symmetric; and they can be structurally rank deficient, i.e., any rank deficiency is evident from the zero/nonzero structure of A. This paper illustrates nicely how graph theory can contribute to improving the efficiency of sparse matrix methods. The paper should appeal to those who need to solve large sparse linear systems, as well as those interested in graph theory. The second paper, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization” by Benjamin Recht, Maryam Fazel, and Pablo Parrilo, has applications to model reduction and system identification, machine learning, and image compression, just to name a few. Given a set of affine constraints for a matrix, the problem is to find a matrix of minimum rank that satisfies these constraints. The authors attack this hard nonconvex optimization problem by replacing it with a convex approximation: Instead of minimizing the rank, they minimize the sum of the singular values, the so-called nuclear norm. Nuclear norm minimization thus extends the compressed sensing framework from finding sparse vectors via $\ell_1$ minimization to finding low-rank matrices via $\ell_1$ minimization of the (vector of) singular values. The purpose of this paper is to justify mathematically why nuclear norm minimization does so well in practice. In addition to extending the restricted isometry property to matrices, the authors also discuss algorithms, computational performance, and numerical issues. This is a fascinating and well-written paper that combines results from compressed sensing, matrix theory, optimization, and probability.

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

Efficient solution of linear matrix equations with application to multistatic antenna array processing

We present a computationally-efficient matrix-vector expression for the solution of a matrix linear least squares problem that arises in multistatic antenna array processing. Our derivation relies on an explicit new relation between Kronecker, Khatri-Rao and Schur-Hadamard matrix products, which involves a selection matrix (i.e., a subset of the columns of a permutation matrix). Moreover, we show that the same selection matrix also relates the vectorization-by-columns operator to the diagonal extraction operator, which plays a central role in our computationally- efficient solution.

On the Hermitian-Generalized Hamiltonian Solutions of Linear Matrix Equations

By means of the properties of Hermitian-generalized Hamiltonian matrices, we give some necessary and sufficient conditions for the solvability of the matrix equation AX = B in Hermitian-generalized Hamiltonian matrix set $HHC^{n\times n}$. In the case where AX = B is solvable in $HHC^{n\times n}$, we derive the general representation of the solutions. We also give the expression of the solution for the corresponding optimal approximation problem.

Solution of linear matrix equations in a *-congruence class

The possible *congruence classes of a square solution to the real or complex linearmatrix equation AX = B are determined. The solution is elementary and self contained, and includesseveral known results as special cases, e.g., X is Hermitian or positive semidefinite, and X is realwith positive definite symmetric part.

Localization of spectrum and stability of certain classes of dynamical systems

We develop a method for the localization of spectra of multiparameter matrix pencils and matrix functions, which reduces the problem to the solution of linear matrix equations and inequalities. We formulate algebraic conditions for the stability of linear systems of differential, difference, and difference-differential equations.