China
Africa
Explore chapters and articles related to this topic
Linear Algebra
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Fatemeh Hamidi Sepehr, Erchin Serpedin
An n × n matrix A = [αk,l] is called stochastic (probabilistic or transition) matrix if all its entries are non-negative (ak,l ≥ 0) and the entries in each row add up to one ∑l=1nak,l=1 A stochastic matrix can be interpreted as the transition probability matrix in a Markov chain, and its use arises in many fields such as probability, statistics, computer science, optimization (majorization theory), etc. A matrix with non-negative entries and with each column summing up to one is also referred to as a stochastic matrix. A doubly stochastic matrix is a matrix with non-negative entries, and in which each row and each column add up to one.
3D Shape Registration Using Spectral Graph Embedding and Probabilistic Matching
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
Avinash Sharma, Radu Horaud, Diana Mateus
(Birkhoff) A matrixAis a doubly stochastic matrix if and only if for some N < ∞ there are permutation matricesP1,…, PNand positive scalars s1,…, sN such that s1 + … + sN = 1 andA = s1P1 + … + sNPN.
Combinatorial Matrix Theory
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Doubly stochastic matrices (see Chapter 9.4) are widely studied because of their connection with probability theory, as every doubly stochastic matrix is the transition matrix of a Markov chain. The reader is referred to Chapter 28 for graph terminology.
A community partitioning algorithm based on network enhancement
Published in Connection Science, 2021
Junjie Hu, Zhanquan Wang, Jiequan Chen, Yonghui Dai
Stochastic matrix is a matrix used to describe the transformation of a Markov chain, each term of which is a non-negative real number representing the probability. If a matrix s a doubly stochastic matrix (Wang et al., 2016), the following conditions shall be met: These two conditions guarantee that every term in a doubly stochastic matrix is a non-negative probability value and that the sum of every row and column is 1. Symmetric doubly stochastic matrix is a symmetric constraint added on the basis of the doubly stochastic matrix, namely , where represents the transpose of the matrix . Symmetric doubly stochastic matrix has the following properties: It has the maximum eigenvalue 1, and the corresponding eigenvector is All eigenvalues are non-negativeIt has n linearly independent eigenvectors
On the convergence of exact distributed generalisation and acceleration algorithm for convex optimisation
Published in International Journal of Systems Science, 2020
Huqiang Cheng, Huaqing Li, Zheng Wang
For , subtracting on the both sides of (23), we obtain Using the fact that and subtracting at the right-hand side of the above equality, we have where the first term uses the fact that . We next bound (30) as where the second inequality follows the fact that . Since W is a doubly stochastic matrix, there holds that . Recalling , we have Applying Lemma 3.3 on the above inequality, the desired result follows.
Adaptive output tracking of linear uncertain discrete-time networked systems over switching topologies
Published in International Journal of Control, 2020
Compared to the distributed observer given in Yan and Huang (2017a), where only the case of undirected jointly connected graphs are considered, our output-based distributed observer is capable to handle a more general class of switching digraphs. In particular, Assumption 3.1 with is always satisfied for the case where all the digraphs are balanced. To illustrate this point, for each , let be the subgraph of in which and is obtained from by removing all edges incident on node 0, and be the matrix consisting of the last N rows and the last N columns of . Then it can be seen that Assumption 3.1 with is satisfied if the digraphs , are balanced. To see this, note that is a doubly stochastic matrix if is balanced. Thus, is a stochastic matrix. Since is the lower right submatrix of in view of (16), is substochastic. Hence, . As a result, (15) is satisfied with , and .