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The theory of matrices and determinants
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
Matrices are used to solve problems in electronics, optics, quantum mechanics, statics, robotics, linear programming, optimisation, genetics and much more. Matrix calculus is a mathematical tool used in connection with linear equations, linear transformations, systems of differential equations and so on, and is vital for calculating forces, vectors, tensions, masses, loads and a lot of other factors that must be accounted for in engineering to ensure safe and resource-efficient structure. Electrical and mechanical engineers, chemists, biologists and scientists all need knowledge of matrices to solve problems. In computer graphics, matrices are used to project a three-dimensional image on to a two-dimensional screen, and to create realistic motion. Matrices are therefore very important in solving engineering problems.
The theory of matrices and determinants
Published in John Bird, Bird's Engineering Mathematics, 2021
Matrices are used to solve problems in electronics, optics, quantum mechanics, statics, robotics, linear programming, optimisation, genetics and much more. Matrix calculus is a mathematical tool used in connection with linear equations, linear transformations, systems of differential equations and so on, and is vital for calculating forces, vectors, tensions, masses, loads and a lot of other factors that must be accounted for in engineering to ensure safe and resource-efficient structure. Electrical and mechanical engineers, chemists, biologists and scientists all need knowledge of matrices to solve problems. In computer graphics, matrices are used to project a 3-dimensional image on to a 2-dimentional screen, and to create realistic motion. Matrices are therefore very important in solving engineering problems.
The theory of matrices and determinants
Published in John Bird, Engineering Mathematics, 2017
Matrices are used to solve problems in electronics, optics, quantum mechanics, statics, robotics, linear programming, optimisation, genetics, and much more. Matrix calculus is a mathematical tool used in connection with linear equations, linear transformations, systems of differential equations and so on, and is vital for calculating forces, vectors, tensions, masses, loads and a lot of other factors that must be accounted for in engineering to ensure safe and resource-efficient structure. Electrical and mechanical engineers, chemists, biologists and scientists all need knowledge of matrices to solve problems. In computer graphics, matrices are used to project a 3-dimensional image on to a 2-dimentional screen, and to create realistic motion. Matrices are therefore very important in solving engineering problems.
Global risk assessment for development processes: from framework to simulation
Published in International Journal of Production Research, 2022
Jelena Petronijevic, Alain Etienne, Ali Siadat
Initial values of risk factors form the input vector (1). Influences between risk factors are captured in the matrix (2). The matrix calculus is performed in several iterations until system either reaches stable, repetitive or chaos state. The chaos state means that the relationships in the map are forming unstable system which is always provoking different output. In risk management, stable states are preferred. The limit of 100 iterations is considered experimentally sufficient for the system to reach stable state if there is any. To calculate and keep the vector values within the boundaries of 0 and 1, the following formula is used: where represents the value of risk factor in the current iteration; is the value of corresponding weight; value of the risk factor currently involved in the calculation and value of observed risk factor in the previous iteration.
Adaptive region reaching control of fully actuated ocean surface vessels
Published in Journal of Control and Decision, 2021
X. M. Sun, S. S. Ge, Q. Xu, Y. Zhou, X. W. Zheng
Step (2) Differentiating with respect to time yields Consider the following Lyapunov function candidate: where is an adaptation gain matrix, and . The time derivative of is given by From theory of the matrix calculus, we get Since , simplifying (22), reduces to We can construct the real control τ and the adaptive updating law as follows: By using (24) and (25), the derivative of becomes where and are positive constants.
Toeplitz matrices for LTI systems, an illustration of their application to Wiener filters and estimators
Published in International Journal of Systems Science, 2018
The sufficiency is easily verified by differentiating (29) with respect to F which by matrix calculus (Gentle, 2007) gives where ⊗ is the Kronecker product of matrices. Equation (32) always gives a positive definite result since LDLT is a Sylvester matrix and is always full rank. The minimum mean-square error is shown in Appendix 1. Note in (31) that there are no z operator terms in the solution at all even though the solution represents an FIR transfer function filter. Usually, a matrix solution like this would be state-space in form but it is also not of that form either. The Toeplitz method enables for us to work in transfer-function mode but without using the z operator.