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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.
Preliminaries on Trusses
Published in A.I. Rusakov, Fundamentals of Structural Mechanics, Dynamics, and Stability, 2020
A square matrix with a zero determinant is referred to as a singular matrix; otherwise a square matrix is called nonsingular. The determinant of matrix A is denoted as det A. The existence and uniqueness of the solution of a linear equation set is defined by whether the coefficient matrix is singular or not. The solution of the linear equation set in vector form (7I.3) exists and is unique on the obligatory condition: det A ≠ 0. It follows that for an absolute term vector b = 0 and upon the condition det A ≠ 0, the solution of equation (7I.3) is only and trivial: x = 0. If det A = 0, the solution may not exist, but if it exists, then there exists an infinite set of other solutions. Thus, for absolute term vector b = 0 upon condition det A = 0, one can find the nontrivial solution x ≠ 0.
Matrix Algebra
Published in Prem K. Kythe, Elements of Concave Analysis and Applications, 2018
The determinant |A| is sometimes denoted by det(A) $ \text{ det}(\mathbf{A}) $ . It is a number or a scalar and is obtained only for square matrices. If |A|=0 $ |\mathbf{A}|=0 $ , then the determinant is said to vanish and the matrix A $ \mathbf{A} $ is said to be singular. A singular matrix is one in which there exists a linear dependence between at least two rows or columns. If |A|≠0 $ |\mathbf{A}|\ne 0 $ , then the matrix A $ \mathbf{A} $ is nonsingular and all its rows and columns are linearly independent.
A fully probabilistic design for stochastic systems with input delay
Published in International Journal of Control, 2021
As can be seen from the above discussion, is, in general, a non-square matrix. Therefore, does not itself have a true inverse. Thus, the introduction of the pseudo-inverse matrix in the optimisation process is necessary. On the other hand, as can be seen from Equation (15), the pseudo-inverse matrix does have the property that where I is the identity matrix. However, note that in general. If the matrix is singular then Equation (13) does not have a unique solution. In this case, if the pseudo-inverse is defined as then, the limit can be shown to always exist and that the limiting value guarantees the optimal solution of Equation (13).
Computer integrated work-space quality improvement of the C4 parallel robot CMM based on kinematic error model for using in intelligent measuring
Published in International Journal of Computer Integrated Manufacturing, 2021
Mohammad Aliakbari, Mehran Mahboubkhah, Mohammadali Sadaghian, Ahmad Barari, Sina Akhbari
When determinants of the matrices of and are equal to zero, they are singular. Therefore, in singularity analysis of a robot, the determinant of these matrices in different poses is calculated and the singular points are obtained. It is a well-established fact that in practice, a tolerance of ε is assumed for the determinant of the matrices instead of zero as shown in Equation (18). If the determinant of a point is equal to or less than , the system registers the pose as a singular pose.