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In this paper, we discuss the following two minimum rank matrix approximation problems in the spectral norm: (i) For given A = A H ∈ C m × m , B ∈ C m × n , determining X ∈ S 1 , such that rank ( X ) = min Y ∈ S 1 rank ( Y ) , S 1 = { Y = Y H ∈ C n × n : ‖ A − B Y B H ‖ 2 = min } . (ii) For given A = − A H ∈ C m × m , B ∈ C m × n , determining X ∈ S 2 , such that rank ( X ) = min Y ∈ S 2 rank ( Y ) , S 2 = { Y = − Y H ∈ C n × n : ‖ A − B Y B H ‖ 2 = min } . By applying the norm-preserving dilation theorem, the Hermitian-type (skew-Hermitian-type) generalized singular value decomposition (HGSVD, SHGSVD), we characterize the expressions of the minimum rank and derive a general form of minimum rank (skew) Hermitian solutions to the matrix approximation problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following minimization problem:where,,,andare given. An efficient inequality relaxation technique is presented to relax the matrix inequality constraint so that there is an optimal solution which is(R,S)-symmetric that minimize, and also satisfies the corrected matrix inequality constraint. A hybrid algorithm with convergence analysis is given to solve this problem. Numerical examples show that the algorithm requires less CPU times when compared with some other methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of factorizing the n × n matrix J n of all ones into the n × n binary matrices. We show that under some conditions on the factors, these are isomorphic to a row permutation of a De Bruijn matrix. Moreover, we consider in particular the binary roots of J n , i.e. the binary solutions to A m = J n . On the one hand, we prove that any binary root with minimum rank is isomorphic to a row permutation of a De Bruijn matrix whose row permutation is represented by a block diagonal matrix. On the other hand, we partially solve Hoffman's open problem of characterizing the binary solutions to A 2 = J n by providing a characterization of the binary solutions to A 2 = J n with minimum rank. Finally, we provide a class of roots which are isomorphic to a De Bruijn matrix. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to solve the problem of finding a geodesic that best fits a given set of time-labelled points on the Grassmann manifold. To achieve this goal, we first derive a very useful simplified formula for the geodesic arc joining two points on the Grassmannian depending explicitly only on the given points. This allows to simplify the expression for the geodesic distance, which is crucial to generalize the fitting problem, and is also used to obtain a simpler characterization of the geometric mean of a finite set of points lying on the Grassmannian, where the given points enter explicitly. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how to use the Gauss map to produce efficient triangulations of parametric surfaces. The central idea is to invert canonical triangulations of the sphere using the Gauss map. The main problem is that the inverse of the Gauss map is not in general well defined. We show how to overcome this problem and formulate an algorithm to generate the desired triangulation. [ABSTRACT FROM PUBLISHER]

In this paper, we present an iterative algorithm for solving the following coupled Sylvester-transpose matrix equationsover the generalized centro-symmetric matrix group (X1,X2, …,Xq). The solvability of the problem can be determined by the proposed algorithm, automatically. If the coupled Sylvester-transpose matrix equations are consistent over the generalized centro-symmetric matrices, then a generalized centro-symmetric solution group can be obtained within finite iterative steps for any initial generalized centro-symmetric matrix group in the exact arithmetic. Furthermore, it is shown that the least-norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be computed by choosing an appropriate initial iterative matrix group. Moreover, the optimal approximate generalized centro-symmetric solution group to a given arbitrary matrix group (V1,V2, …,Vq) can be derived by finding the least-norm generalized centro-symmetric solution group of a new coupled Sylvester-transpose matrix equations. Finally, some numerical results are given to illustrate the validity and practicability of the theoretical results established in this work. [ABSTRACT FROM PUBLISHER]

In this paper, an iterative method is presented to solve the following constrained minimum Frobenius norm residual problem: [image omitted] where [image omitted] is a linear operator from Rm×n onto [image omitted] , [image omitted] , [image omitted] is a linear self-conjugate involution operator. By this method, for any initial matrix [image omitted] , a solution can be obtained in finite iteration steps in the absence of roundoff errors. The least norm solution can be derived when an appropriate initial matrix is chosen. In addition, the optimal approximation solution in the solution set of the above problem to a given matrix can also be derived by this method. Several numerical examples are given to show the efficiency of the proposed iterative method. [ABSTRACT FROM AUTHOR]