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Linear Systems
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
To find a way to get a (near) LU decomposition of matrices that require pivoting–which, because of stability concerns, is the typical case, unlike the specially prepared examples in the previous section–let's revisit Gaussian elimination. We perform Gaussian elimination using two of the three elementary row operations. Each elementary row operation corresponds to an elementary matrix that implements that operation by premultiplication; this matrix is always the matrix that arises by applying the elementary row operation to the identity matrix. That is, if A is an n×n matrix then performing Ri↔Rj to A is equivalent to forming the matrix product PA where P is the matrix obtained from In by interchanging rows i and j of it, and similarly for the other row operations.
Introduction to Linear Algebra
Published in Timothy Bower, ®, 2023
Note that each elementary matrix only differs from the identity matrix at one element. The inverse of each elementary matrix is found by just changing the sign of the additional nonzero value to the identity matrix.
Long-time behaviour of solutions of superlinear systems of differential equations
Published in Dynamical Systems, 2023
For , let be the elementary matrix , where and are the Kronecker delta symbols. For , define Then one immediately has Thanks to (114), each is a projection, and is the eigenspace of A associated with the eigenvalue .
Load distribution and radial deformations for planetary roller screw mechanism with axial load, radial load and turning torque
Published in Mechanics Based Design of Structures and Machines, 2022
Jiacheng Miao, Xing Du, Chaoyang Li, Bingkui Chen
Where the stiffness matrices of screw and roller are KS and KR, respectively, and can be assembled by beam element Kbeam. The stiffness matrix of nut KN can be constructed using the elementary matrix Kbar of a bar element. Moreover, after adding the elementary contact stiffness KS,rtx, KS,rty, KS,rtz, KN,rtx, KN,rty, KN,rtz to the non-linear spring matrix Ksc and Knc, and adding thread stiffness KMTix, KMTiy, KMTiz, KNTix, KNTiy, KNTiz to the linear spring matrix KT, the non-linear matrix Kn of the MR-BBS system with multi-springs can be established.
Convergence characteristics of iterative learning control for discrete-time singular systems
Published in International Journal of Systems Science, 2021
Ijaz Hussain, Xiaoe Ruan, Yan Liu
Because is an elementary matrix whose the first column up to the column are arrayed as the , the , up to the and then the , the , up to the columns of the unity matrix , by the attribute of the elementary matrix, it is assertive that the columns of the matrix may be rearranged as the same as the above-mentioned array when it is right-multiplied by . In a similar way, the rows of matrix may be rearranged when it is left-multiplied by .