Rank structure of generalized inverses of rectangular banded matrices*

Abstract We describe rank structures in generalized inverses of possibly rectangular banded matrices. In particular, we show that various kinds of generalized inverses of rectangular banded matrices have submatrices whose rank depends on the bandwidth and on the nullity of the matrix. Moreover, we give an explicit representation formula for some generalized inverses of strictly banded matrices.     

A bibliography on semiseparable matrices*

Abstract Currently there is a growing interest in semiseparable matrices and generalized semiseparable matrices. To gain an appreciation of the historical evolution of this concept, we present in this paper an extensive list of publications related to the field of semiseparable matrices. It is interesting to see that semiseparable matrices were investigated in different fields, e.g., integral equations, statistics, vibrational analysis, independently of each other. Also notable is the fact that leading statisticians at that time used semiseparable matrices without knowing their inverses to be tridiagonal. During this historical evolution the definition of semiseparable matrices has always been a difficult point leading to misunderstandings; sometimes they were defined as the inverses of irreducible tridiagonal matrices leading to generator representable matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal. In this overview we present a list of interesting results which contributed to the evolution of the concept of semiseparable matrices as we now know them.     

On some ways of approximating inverses of banded matrices in connection with deriving preconditioners based on incomplete block factorizations

A unified approach of deriving band approximate inverses of band symmetric positive definite matrices is considered. Such band approximations to the inverses of successive Schur complements are required throughout incomplete block factorizations of block-tridiagonal matrices. Such block-tridiagonal matrices arise, for example, in finite element solution of second order elliptic differential equations. A sharp decay rate estimate for inverses of blocktridiagonal symmetric positive definite matrices is given in addition. Numerical tests on a number of model elliptic boundary value problems are presented comparing thus derived preconditioning matrices.     

Report on test matrices for generalized inverses

This paper is a comprehensive report on test matrices for the generalized inversion of matrices. Two principles are described how to construct singular square or arbitrary rectangular test matrices and their Moore-Penrose inverses. By prescribing the singular values of the matrices or by suitably choosing the free parameters test matrices with condition numbers of any size can be obtained. We also deal with test matrices which are equal to their Moore-Penrose inverse. In addition to many advices how to construct test matrices the paper presents many test matrices explicitly, in particular singular square matrices of ordern, sets of 7×6 and 7×5 matrices of different rank, a set of 5×5 matrices which are equal to their Moore-Penrose inverse and some special test matrices known from literature. For the set of 7×6 parameter matrices also the singular values corresponding to six values of the parameter are listed. For three simple parameter matrices of order 5×4 and 6×5 even test results obtained by eight different algorithms are quoted. As “by-products” the paper contains inequalities between condition numbers of different norms, representations for unitary, orthogonal, column-orthogonal and row-orthogonal matrices, a generalization of Hadamard matrices and representations of matrices which are equal to their Moore-Penrose inverse (or their inverse). All test matrices given in this paper may also be used for testing algorithms solving linear least squares problems.     

Comments on “An analytical algorithm for generalized low-rank approximations of matrices”

Low rank approximations of matrices have been widely used in pattern recognition and machine learning. Based on a sequence of matrices, a generalized low rank approximation problem was presented and an iterative scheme was given by Liang and Shi recently proposed an analytical scheme for this approximation problem. In this paper, we identify the weakness in their scheme and prove that their algorithm is incorrect.     

The representations and computations of generalized inverses $$A^{(1)}_{T,S}$$, $$A^{(1,2)}_{T,S}$$AT,S(1,2) and the group inverse

In this paper, we derive novel representations of generalized inverses \(A^{(1)}_{T,S}\) and \(A^{(1,2)}_{T,S}\), which are much simpler than those introduced in Ben-Israel and Greville (Generalized inverses: theory and applications. Springer, New York, 2003). When \(A^{(1,2)}_{T,S}\) is applied to matrices of index one, a simple representation for the group inverse \(A_{g}\) is derived. Based on these representations, we derive various algorithms for computing \(A^{(1)}_{T,S}\), \(A^{(1,2)}_{T,S}\) and \(A_{g}\), respectively. Moreover, our methods can be achieved through Gauss–Jordan elimination and complexity analysis indicates that our method for computing the group inverse \(A_{g}\) is more efficient than the other existing methods in the literature for a large class of problems in the computational complexity sense. Finally, numerical experiments show that our method for the group inverse \(A_{g}\) has highest accuracy among all the existing methods in the literature and also has the lowest cost of CPU time when applied to symmetric matrices or matrices with high rank or small size matrices with low rank in practice.     

Generalized Bezoutian and Sylvester matrices in multivariable linear control

Generalized Bezoutian and Sylvester matrices are defined and discussed in this short paper. The relationship between these two forms of matrices is established. It is shown that the McMillan degree of a real rational function can be ascertained by checking the rank of either one of these generalized matrices formed using a polynomial matrix fraction decomposition of the prescribed transfer function matrix. Earlier established results by Rowe and Munro are obtained as a special case. Several theorems related to the rank testing and other properties of the generalized matrices are discussed and various research problems are listed in the conclusion.     

$$\mathcal $$-FAINV: hierarchically factored approximate inverse preconditioners

Given a sparse matrix, its LU-factors, inverse and inverse factors typically suffer from substantial fill-in, leading to non-optimal complexities in their computation as well as their storage. In the past, several computationally efficient methods have been developed to compute approximations to these otherwise rather dense matrices. Many of these approaches are based on approximations through sparse matrices, leading to well-known ILU, sparse approximate inverse or factored sparse approximate inverse techniques and their variants. A different approximation approach is based on blockwise low rank approximations and is realized, for example, through hierarchical (\(\mathcal H\)-) matrices. While \(\mathcal H\)-inverses and \(\mathcal H\)-LU factors have been discussed in the literature, this paper will consider the construction of an approximation of the factored inverse through \(\mathcal H\)-matrices (\(\mathcal H\)-FAINV). We will describe a blockwise approach that permits to replace (exact) matrix arithmetic through approximate efficient \(\mathcal H\)-arithmetic. We conclude with numerical results in which we use approximate factored inverses as preconditioners in the iterative solution of the discretized convection–diffusion problem.     

Decoding of random network codes

We consider the decoding for Silva-Kschischang-Kötter random network codes based on Gabidulin’s rank-metric codes. The model of a random network coding channel can be reduced to transmitting matrices of a rank code through a channel introducing three types of additive errors. The first type is called random rank errors. To describe other types, the notions of generalized row erasures and generalized column erasures are introduced. An algorithm for simultaneous correction of rank errors and generalized erasures is presented. An example is given.     

Design of a minimal-order observer for singular systems

This paper considers the design of an observer for generalized state-space systems using the concept of generalized inverses. In contrast to the Luenberger observer theory, a new method is proposed which does not presuppose the observer structure. It is shown that under certain conditions it is possible to construct a minimal-order observer for this class of systems. Illustrative examples are included.     

An algorithm for inverting rational matrices

We propose an algorithm for computing the inverses of rational matrices and in particular the inverses of polynomial matrices. The algorithm is based on minimal state space realizations of proper rational matrices and the matrix inverse lemma and is implemented as a MATLAB¹ function. Experiments show that the algorithm gives accurate results for typical rational matrices that arise in analysis and design of linear multivariable control systems. Illustrative examples are given.     

Multivariable system design by transfer function synthesis

This paper is concerned with the problem of designing a constant output feedback matrix to synthesize the denominator and certain numerator polynomials of the transfer function matrix of a linear multivariable system. The problem is tackled by first using the special case of unity rank feedback and then removing this restriction to cover the case of unconstrained feedback.

Simple expressions relating closed-loop and open-loop transfer functions of multi-variable systems and unity rank feedback are first established. These are then used to develop a recursive synthesis algorithm employing unity rank output feedback. The algorithm uses the pseudo-inverse concept for obtaining least-squares solutions of sots of simultaneous linear equations. The extension from unity rank feedback to unrestricted rank feedback is carried out by considering the feedback matrix as a sum of unity rank matrices. Using this concept and the expressions derived earlier for unity rank feedback, a recursive algorithm is developed for calculating unrestricted rank output feedback matrices. Examples are given to illustrate the synthesis procedure using both unity rank and unrestricted rank feedback matrices.     

Pole assignment using matrix generalized inverses

The problem of pole assignment in a completely controllable linear time-invariant system dx/t = Ax + Bu, y = Cx is considered. A method using matrix generalized inverses is developed for the computation of a matrix K such that the matrix A + BK has prescribed eigenvalues which need satisfy only the condition that a certain number of them are distinct and real; then a feedback law of the form u = r + Kx can be used to achieve the desired pole-placement. The method does not require solution of sets of non-linear equations or manipulation of polynomial matrices, and no knowledge of eigenvalues and/or eigenvectors of A is necessary. If the computed matrix K and the given matrix C satisfy a consistency condition, a matrix Kν such that KνC = K can be directly obtained from K and the desired pole-placement can be realized by an output feedback law u = r + Kνy.     

基于广义轮换矩阵的伪随机广义二进制轮换矩阵设计

压缩感知中,测量矩阵在信号的获取和重构过程中起着重要的作用.传统的随机测量矩阵在采样率较高的情况下,能够获得比较好的重构效果,但在低采样率下的重构效果不够理想.确定性测量矩阵自身存在一些限制因素,与随机测量矩阵相比,重构效果有所降低.基于广义轮换矩阵(GR),提出了两种结构随机矩阵:广义二进制轮换矩阵(GBR)和伪随机广义二进制轮换矩阵(PGBR).仿真结果表明,相对于传统的测量矩阵,新的测量矩阵在二维图像重建方面效果较好,所需重构时间相差不大,在较低的采样率下能够获得更加精确的重建.     

鲁棒分散控制系统闭环传递矩阵的界

本文将Nyquist阵列用于对象模型具有非结构不确定性的分散控制系统.在假定返回差 矩阵为广义块对角优势的条件下,得到了当某个子块反馈为开路(或某个反馈回路为开路)时 的闭环传递矩阵(或闭环传递函数)的界及其特征值的包含域.这是标准的Ostrowski带的再 推广.     

酉对称矩阵的QR分解及其算法

该文讨论了酉对称矩阵QR分解中Q矩阵和R矩阵与母矩阵的Q矩阵和R矩阵之间的定量关系．从矩阵正交相抵的概念出发,给出了矩阵酉相抵的概念,证明了酉对称矩阵与母矩阵之间的酉相抵性,得到了酉相抵矩阵的Moore—Penrose逆等一些新的结论．同时,给出了酉对称矩阵的QR分解及其Moore—Penrose逆矩阵的算法．     

Finite transmission zeros of linear time-invariant generalized state space systems

In this paper, the problem of finding inputs which generate zero outputs in linear time-invariant control systems of the form E(dx/dt) = Ax + Bu is considered where E and A are not necessarily square matrices. This problem is solved using the theory of matrix generalized inverses.     

基于字典和加权低秩恢复的显著目标检测

显著目标检测旨在辨别出自然图像中的显著区域。为了提高检测效果,提出了基于字典和加权低秩恢复的显著目标检测。首先,在低秩恢复模型中融入字典,以更好地将低秩矩阵和稀疏矩阵分离;然后,获取颜色、位置和边界连接先验对应的稀疏矩阵,根据其显著值生成先验系数;最后,将3个先验用自适应系数组合的方式构造权重矩阵,并融入到低秩恢复模型中。在4个具有挑战性的数据集上将其与11种算法进行比较,实验结果表明,所提算法的效果最好。     

Algebraic algorithms for least squares problem in quaternionic quantum theory

Quaternionic least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations AX≈B that is appropriate when there is error in the matrix B. In this paper, by means of complex representation of a quaternion matrix, we introduce a concept of norm of quaternion matrices, discuss singular values and generalized inverses of a quaternion matrix, study the QLS problem and derive two algebraic methods for finding solutions of the QLS problem in quaternionic quantum theory.