Some Natures on Matrix Frobenius Norm

2010 ◽  
Author(s):  
Yang XingDong ◽  
Ding ZhiYing
Keyword(s):  
Frobenius Norm

scholarly journals The core inverse and constrained matrix approximation problem

2020 ◽  
Vol 18 (1) ◽  
pp. 653-661 ◽  
Author(s):  
Hongxing Wang ◽  
Xiaoyan Zhang
Keyword(s):  
Unique Solution ◽  
Approximation Problem ◽  
Frobenius Norm ◽  
Matrix Approximation ◽  
The Core ◽  
Core Inverse

Abstract In this article, we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse: ||Mx-b|{|}_{F}=\hspace{.25em}\min \hspace{1em}\text{subject}\hspace{.25em}\text{to}\hspace{1em}x\in {\mathcal R} (M), where M\in {{\mathbb{C}}}_{n}^{\text{CM}} . We get the unique solution to the problem, provide two Cramer’s rules for the unique solution and establish two new expressions for the core inverse.

scholarly journals Modified covariance Frobenius norm detector for cognitive radio systems

2011 ◽  
Vol 8 (10) ◽  
pp. 762-766 ◽  
Author(s):  
Ming Jin ◽  
Youming Li ◽  
Qi Zeng
Keyword(s):  
Cognitive Radio ◽  
Frobenius Norm ◽  
Cognitive Radio Systems ◽  
Radio Systems

scholarly journals COMMUTATORS OF SKEW-SYMMETRIC MATRICES

2005 ◽  
Vol 15 (03) ◽  
pp. 793-801 ◽  
Author(s):  
ANTHONY M. BLOCH ◽  
ARIEH ISERLES
Keyword(s):  
Optimal Control ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Geometric Integration ◽  
Symmetric Matrices ◽  
Frobenius Norm

In this paper we develop a theory for analysing the "radius" of the Lie algebra of a matrix Lie group, which is a measure of the size of its commutators. Complete details are given for the Lie algebra 𝔰𝔬(n) of skew symmetric matrices where we prove [Formula: see text], X, Y ∈ 𝔰𝔬(n), for the Frobenius norm. We indicate how these ideas might be extended to other matrix Lie algebras. We discuss why these ideas are of interest in applications such as geometric integration and optimal control.

scholarly journals A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Xuefeng Duan ◽  
Chunmei Li
Keyword(s):  
Iterative Algorithm ◽  
Projection Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Von Neumann ◽  
Numerical Examples ◽  
Matrix Nearness Problem ◽  
Least Frobenius Norm Solution ◽  
Alternating Projection ◽  
The Matrix

Based on the alternating projection algorithm, which was proposed by Von Neumann to treat the problem of finding the projection of a given point onto the intersection of two closed subspaces, we propose a new iterative algorithm to solve the matrix nearness problem associated with the matrix equations AXB=E, CXD=F, which arises frequently in experimental design. If we choose the initial iterative matrix X0=0, the least Frobenius norm solution of these matrix equations is obtained. Numerical examples show that the new algorithm is feasible and effective.

scholarly journals A Gradient System for Low Rank Matrix Completion

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 51 ◽  
Author(s):  
Carmela Scalone ◽  
Nicola Guglielmi
Keyword(s):  
Sparse Matrix ◽  
Matrix Completion ◽  
Low Rank ◽  
Frobenius Norm ◽  
Gradient System ◽  
Large Matrix ◽  
Rank Matrix ◽  
Matrix Differential Equations ◽  
Low Rank Matrix Completion ◽  
Matrix Differential

In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature.

scholarly journals An Iterative Algorithm for the Reflexive Solution of the General Coupled Matrix Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-15
Author(s):  
Zhongli Zhou ◽  
Guangxin Huang
Keyword(s):  
Iterative Algorithm ◽  
Matrix Group ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Matrix Solution ◽  
Initial Matrix ◽  
Special Cases ◽  
Reflexive Matrix ◽  
General Coupled Matrix Equations ◽  
Sylvester Matrix Equations

The general coupled matrix equations (including the generalized coupled Sylvester matrix equations as special cases) have numerous applications in control and system theory. In this paper, an iterative algorithm is constructed to solve the general coupled matrix equations over reflexive matrix solution. When the general coupled matrix equations are consistent over reflexive matrices, the reflexive solution can be determined automatically by the iterative algorithm within finite iterative steps in the absence of round-off errors. The least Frobenius norm reflexive solution of the general coupled matrix equations can be derived when an appropriate initial matrix is chosen. Furthermore, the unique optimal approximation reflexive solution to a given matrix group in Frobenius norm can be derived by finding the least-norm reflexive solution of the corresponding general coupled matrix equations. A numerical example is given to illustrate the effectiveness of the proposed iterative algorithm.

scholarly journals The 1995-2018 global evolution of the network of amicable and hostile relations among nation-states

2020 ◽  
Vol 3 (1) ◽  
Author(s):  
Omid Askarisichani ◽  
Ambuj K. Singh ◽  
Francesco Bullo ◽  
Noah E. Friedkin
Keyword(s):  
Large Scale ◽  
Transition Probabilities ◽  
Balance Theory ◽  
Frobenius Norm ◽  
Structural Balance ◽  
Interdisciplinary Field ◽  
Global Evolution ◽  
Nation States ◽  
Appraisal Systems ◽  
Large Scale Networks

AbstractThere has been longstanding interest in the evolution of positive and negative relationships among countries. An interdisciplinary field of study, Structural Balance Theory, has developed on the dynamics of such appraisal systems. However, the advancement of research in the field has been impeded by the lack of longitudinal empirical data on large-scale networks. We construct the networks of international amicable and hostile relations occurring in specific time-periods in order to study the global evolution of the network of such international appraisals. Here we present an empirical evidence on the alignment of Structural Balance Theory with the evolution of the structure of this network, and a model of the probabilistic micro-dynamics of the alterations of international appraisals during the period 1995-2018. Also remarkably, we find that the trajectory of the Frobenius norm of sequential transition probabilities, which govern the evolution of international appraisals among nations, dramatically stabilizes.

scholarly journals An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Iterative Algorithm ◽  
Special Kind ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Sylvester Matrix ◽  
Special Cases ◽  
The Matrix ◽  
Solution Pair ◽  
Sylvester Matrix Equations ◽  
Matrix Pair

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations(AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matricesXandY. When the matrix equations are consistent, for any initial generalized reflexive matrix pair[X1,Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair[X̂,Ŷ]to a given matrix pair[X0,Y0]in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair[X̃*,Ỹ*]of a new corresponding generalized coupled Sylvester matrix equation pair(AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), whereM̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

Two-Stage Robust Tracking Controller for Linear Systems With Known Uncertainty Using Filtered Basis Functions

2020 ◽  
Author(s):  
Keval S. Ramani ◽  
Chinedum E. Okwudire
Keyword(s):  
Linear Combination ◽  
Linear Systems ◽  
Tracking Error ◽  
Basis Functions ◽  
Frobenius Norm ◽  
Control Input ◽  
Two Stage ◽  
Nominal Model ◽  
Control Of Linear Systems ◽  
Optimal Set

Abstract There is growing interest in the use of the filtered basis functions (FBF) approach to track linear systems, especially nonminimum phase (NMP) plants, because of the distinct advantages it presents as compared to other popular methods in the literature. The FBF approach expresses the control input to the plant as a linear combination of basis functions. The basis functions are forward filtered through the plant dynamics and the coefficients of the linear combination are selected such that the tracking error is minimized. This paper proposes a two-stage robust filtered basis functions approach for tracking control of linear systems in the presence of known uncertainty. In the first stage, the nominal model for filtering the basis functions is selected such that a Frobenius norm metric which considers the known uncertainty is minimized. In the second stage, an optimal set of basis functions is selected such that the effect of uncertainty is minimized for the nominal model selected in the first stage. Experiments on a 3D printer, demonstrate up to 7 times improvement in tracking performance using the proposed method as compared to the standard FBF approach.

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