China
Africa
Explore chapters and articles related to this topic
Partitioned Matrices
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
The matrix A ∈ Fm×n is said to be partitioned into the submatrices A[αi, βj], 1 ≤ i ≤ r, 1 ≤ j ≤ s. A block matrix is a matrix that is partitioned into submatrices A[αi, βj] with the row indices {1, …, m} and column indices {1, …, n} partitioned into subsets sequentially, i.e., α1 = {1, …, i1}, α2 = {i1 + 1, …, i2}, etc.
Linear Algebra
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Fatemeh Hamidi Sepehr, Erchin Serpedin
A matrix A is referred to as a block matrix if it can be represented in terms of sub-matrices Ai,j of appropriate sizes as follows: () A=[A1,1A1,2…A1,nA2,1A2,2…A2,n⋮⋮⋱⋮Am,1Am,2…Am,n].
Solving systems of algebraic equations
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
A block matrix may be viewed as a matrix of distinct smaller matrices, typically arrayed on or near the diagonal of the full matrix. They may be encountered, for example, in input-output tables where the inputs fall into discrete clusters. The limSolve package has the function Solve.block that “solves the linear system A*X=B where A is an almost block diagonal matrix of the form: TopBlock
Tracking control for a class of uncertain complex dynamical networks with outgoing links dynamics
Published in International Journal of Control, 2023
Peitao Gao, Yinhe Wang, Juanxia Zhao, LiLi Zhang, Shengping Li
Consider the Lyapunov candidate function given by the following Equation (11). In addition, from Schur Complement theorem (Y. H. Liu et al., 2015; Ma et al., 2011), it is seen that the block-matrix is the positive definition matrix. Differentiating with respect to time and employing the controllers (7) and (8) for NS, we can get Then, substituting the ATO (6) for LS into Equation (12) yields It can be derived from Equation (13) that is the negative semi-definite function, therefore, the closed error systems (9) (about and ) and (10) (about ) are stable, that is, , and are bounded. Therefore, based on Barbalat Lemma (Q. Y. Wang et al., 2008; Y. H. Wang et al., 2012), it can be derived that the tracking errors 0 for NS and 0 for LS hold, meanwhile, the state of NS and the outgoing links vector of LS are bounded, .
Event-based mixed ℋ ∞ and passive filtering for discrete singular stochastic systems
Published in International Journal of Control, 2020
Yingqi Zhang, Peng Shi, Ramesh K. Agarwal, Yan Shi
On the other hand, letting denote the block matrix of the first rows and n columns of and using Schur complement, we can derive from (11a) that which leads to Observe that and the special structure of and , there exist non-singular matrices , and an invertible matrix such that and Note that and denotes the block matrix of the first n rows and n columns of , one can obtain that where ⋆ is not relevant component in the following discussion. Accordingly, pre- and -post multiplying (24) by the block-diagonal matrix and , respectively, one can obtain where . It can be easily derived from (25) that , and furthermore is non-singular, which implies that the error DSJNS (8a) and (8b) is regular and causal. Thus, the proof is completed.
Physically motivated structuring and optimization of neural networks for multi-physics modelling of solid oxide fuel cells
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Andreas Rauh, Julia Kersten, Wiebke Frenkel, Niklas Kruse, Tom Schmidt
where with the identity matrix holds. Assuming , is a block matrix