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Finite Element Concepts In One-Dimensional Space
Published in Steven M. Lepi, Practical Guide to Finite Elements, 2020
Notice that m rows in the A-matrix yields m rows in the resultant; n columns in the /¿-matrix results in n columns in the resultant. The transpose operation specifies that the rows of a matrix are changed into columns, or vice versa. Consider the A-matrix in (2.7.6) and its transpose, denoted as AT:
Linear Algebra
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
An operation that we have seen earlier is the transpose of a matrix. The transpose of a matrix is a new matrix in which the rows and columns are interchanged. If we write an n × m matrix A in standard form as () A=(a11a12⋯a1ma21a22⋯a2m⋮⋮⋱⋮an1an2⋯anm),
Power-Aware Characteristics of Matrix Operations on Multicores
Published in Applied Artificial Intelligence, 2021
Guruprasad Konnurmath, Satyadhyan Chickerur
Matrix Transpose is a building block algorithm for many applications that performs conversion of array of rows M by columns N (i.e. M*N) to array rows N by columns M (i.e. N*M). Whenever the offload of algebraic libraries to GPUs is high, increased performance for in-place transposition is required. Hence, this in-place transpose have to be best fit for most of GPU architectures because of its minimal availability of on-board memory and maximum throughput. The dense matrix transpose (MatTran) preferably designed for memory related manipulations with lesser amount of required computation for memory indices and thread’s ID. To completely utilize GPU’s capability of parallel processing, every multiple rows of matrix are simultaneously interpreted. Intermediate outcomes are recorded in local memory, holding them to write back into global memory.
Investigation of worsted woven fabric’s static friction coefficient considering fabric direction
Published in The Journal of The Textile Institute, 2020
R. Yavari Rameshe, F. Mousazadegan, M. Latifi
A point to be discovered is the impact of changing the samples that exist on the sled and platform with each other on the friction coefficient. Since the entries of a symmetric matrix are symmetric with respect to the main diagonal, if the entries are written as it indicates that according to Equation (1): It is known that in a symmetric matrix, the original matrix is equal to its transpose. This matrix’s property is used in order to evaluate symmetry of friction matrix. On this purpose, we tried to remake the friction coefficient matrix by its transpose. To this end, the diagonal matrix is subtracted from the upper triangular matrix of µ to obtain the off-diagonal elements of the upper triangular matrix as shown in Equation (2): where A = diagonal matrix in which all off-diagonal elements are zero and the diagonal elements are equal to the diagonal elements of µ, B = upper triangular matrix of µ, the transpose of the acquired matrix C presents the transpose of the upper triangular matrix of µ without main diagonal elements (Equation 3). If matrix µ is symmetric, then the sum of the transpose of the matrix C (that is shown by D) and upper triangular matrix of µ will be equal to the original friction matrix (Equation 4). To probe the similarity of the original friction matrix and obtained matrix E, matrix differences is calculated from Equation (5): It is clear that for a symmetric matrix, the differences matrix is zero. However, since the µ is obtained based on experimental friction measurement, some errors are unavoidable. To find a value that can be utilized to decide, norm of the differences matrix and friction matrix is computed as presented in Equation (6): where g is the norm of matrix F and h is the norm of matrix µ, the original friction coefficient matrix
MIMO Nyquist interpretation of the large gain theorem
Published in International Journal of Control, 2020
Ryan James Caverly, Richard Pates, Leila Jasmine Bridgeman, James Richard Forbes
In this paper, boldface letters represent matrices, script letters denote operators, and simple letters denote scalars. The identify matrix is written as and a matrix filled with zeros is written as . Summation points within block diagrams are positive unless otherwise noted. Recall that if , and if , , where for and for t>T. The minimum and maximum singular values of a matrix are and , respectively, where are the singular values of . Using a slight abuse of notation, an by matrix of proper real rational transfer functions is denoted as . The norm of an asymptotically stable transfer matrix is . The complex conjugate transpose of the matrix is . The number of counter-clockwise (CCW) encirclements of the origin made by the Nyquist plot of , also known as the winding number about the origin, is denoted as . The number of closed right-half plane (CRHP) poles of counted according to the Smith-McMillan degree (Skogestad & Postlethwaite, 2007, p. 154) is written as .