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Linear Algebra and Matrices
Published in William F. Ames, George Cain, Y.L. Tong, W. Glenn Steele, Hugh W. Coleman, Richard L. Kautz, Dan M. Frangopol, Paul Norton, Mathematics for Mechanical Engineers, 2022
The n × n identity matrix I has the property that IA = AI = A for every n × n matrix A. If A is square, and if there is a matrix B such at AB = BA = I, then B is called the inverse of A and is denoted A−1. This terminology and notation are justified by the fact that a matrix can have at most one inverse. A matrix having an inverse is said to be invertible, or nonsingular, while a matrix not having an inverse is said to be noninvertible, or singular. The product of two invertible matrices is invertible and, in fact, (AB)−1 = B−1A−1. The sum of two invertible matrices is. obviously, not necessarily invertible.
Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
An n × n matrix A is called singular if there exists a vector x≠0 $ {\mathbf{x}} \ne {\mathbf{0}} $ such that Ax=0 $ A{\mathbf{x}} = {\mathbf{0}} $ or ATx=0 $ A^{T} {\mathbf{x}} = {\mathbf{0}} $ . (Note that x=0 $ {\mathbf{x}} = 0 $ means all components of x are zero). If a matrix is not singular, it is non‐singular.
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
The n×n identity matrix I has the property that IA=AI=A for every n×n matrix A. If A is square, and if there is a matrix B such that AB=BA=I, then B is called the inverse of A and is denoted A-1. This terminology and notation are justified by the fact that a matrix can have at most one inverse. A matrix having an inverse is said to be invertible, or nonsingular, while a matrix not having an inverse is said to be noninvertible, or singular. The product of two invertible matrices is invertible and, in fact, (AB)-1=B-1A-1. The sum of two invertible matrices is, obviously, not necessarily invertible.
Adaptive distributed observer for an uncertain leader with an unknown output over directed acyclic graphs
Published in International Journal of Control, 2021
Since 0 is a known eigenvalue, without loss of generality, we can further assume that the matrix is nonsingular. Thus, Assumption 2.2 is equivalent to saying that the matrix is nonsingular and all of its eigenvalues are semi-simple with zero real part. Note that Assumption 2.2 is also equivalent to saying that the leader's signal is a multi-tone sinusoidal signal.
Application of facial reduction to H ∞ state feedback control problem
Published in International Journal of Control, 2019
One can construct so that the matrix is non-singular since is of full column rank. From these constraints in (18), for any solution in (18), we have