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Matrix Algebra
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
A square matrix is one having the same number of rows as it has columns. The following are examples of square matrices: (23-14)(32045-12-37).
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),
Matrices and determinants
Published in Surinder S. Virdi, Advanced Construction Mathematics, 2019
Each number or symbol in a matrix is known as the element of the matrix. The number of rows and columns in a matrix determines its order, for example, (10497) is a 2 × 2 matrix; the first value is used for the number of rows and the second value used for the number of columns.
Development of a novel integrated value engineering and risk assessment (VENRA) framework for shipyard performance measurement: a case study for an Indonesian shipyard
Published in Ships and Offshore Structures, 2023
Imam Baihaqi, Iraklis Lazakis, Rafet Emek Kurt
Step 4: Obtain the fuzzy total-relation matrix using equations (4) to (7). where, and I is the identity matrix. Identity matrix I is square matrix with ones on the main diagonal and zeros elsewhere.
A Novel Magic Square Based Physical Reconfiguration for Power Enhancement in Larger Size Photovoltaic Array
Published in IETE Journal of Research, 2021
G. Harish Kumar Varma, Venugopal Reddy Barry, Rohit Kumar Jain
Let the order of the matrix is (mxn) where m, n indicates the rows and columns of a matrix. An array of order (6 × 6) is considered for investigation. Figure 2 shows the position of elements in the conventional arrangement of the (6 × 6) PV array.
A methodological approach to assess the territorial vulnerability in terms of people and road characteristics
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2022
Roberta Maletta, Giuseppe Mendicino
The principal component vector indicates essentially a linear combination of the original variables, based on the eigenvector. The last step in performing PCA is to re-arrange the original data with the final principal components. In order to replace the original data axis with the newly formed Principal Components, it is necessary to multiply the transpose of the original data set by the transpose of the obtained feature vector. Then in general, computation of the principal components reduces to the solution of an eigenvalue-eigenvector problem for a positive-semi-definite symmetric matrix. Eigenvectors and eigenvalues provide the eigen-decomposition of a matrix. Specifically, PCA is obtained from the eigen-decomposition of a covariance matrix and from a physical point of view the principal component represents the influence of individual variables. The PCA applied on the matrix gives a realistic indication of the categories of people that characterise the territorial context studied. However, the PCA in this study was also applied on a modified matrix, based on specific weighted parameters wi assigned to each variable xi, to identify intrinsic characteristics of vulnerability classes. It is worth noting that, there is no a unique approach utilised in the literature to correctly identify the weights adopted in the definition of the vulnerability indicators. As a matter of fact, in some cases no weights are assigned, since the variables are considered as independent and equally important (Dwyer et al. 2004). On the other hand, some studies weight indicator values according to perceptions of the importance of certain indicators (Granger and Hayne 2001). Weighting can be very subjective in the absence of adequate data or mathematical methods, but previous studies have found that weights based on experience of the researcher as well as inputs from experts in most cases were better than applying no weights at all (Dwyer et al. 2004; Damm 2010). In the present study, specifically in the input data collection, weights wi were based on a experts’ local knowledge, experience and intuition. Then weights were allocated by using a scale, from 0 to 1 for comparison purposes: 0 corresponds to the lowest vulnerability and 1 to the highest vulnerability. However, in the modelling phase, weights are assigned on the basis of the PCA analysis. As a consequence, the latter PCA quantity can be assumed to be a measure of a people’s vulnerability index, which can be presented in normalised format in the range between 0 and 1 by means of the following normalisation equation: