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A Dynamic Factor and Problem of Finding Natural Frequencies
Published in A.I. Rusakov, Fundamentals of Structural Mechanics, Dynamics, and Stability, 2020
The square matrix A is referred to as diagonalizable if there exists nonsingular matrix S which converts matrix A to a diagonal form by means of relationship: Λ=SAS−1. This definition is applicable both for matrices over the field of complex numbers and for real matrices. Here, we consider only real matrices and real vector spaces. It is known that a matrix of order n is diagonalizable if and only if it has n linearly independent eigenvectors. In matrix analysis, it is proved that the product of two symmetric positive definite matrices is a diagonalizable matrix, and eigenvalues of this product are positive.*
Invertibility
Published in Crista Arangala, Exploring Linear Algebra, 2019
An n × n matrix A is diagonalizable if there exists matrix P such that D = P−1AP, and D is a diagonal matrix. In addition, matrix A is diagonalizable if it has n linearly independent eigenvectors. The n linear independent eigenvectors are the n columns of P. Create a program that determines if any square matrix A is diagonalizable and diagonalizes A if it is diagonalizable.
Matrices for Engineers
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
Definition 26. A square matrix A is said to be diagonalizable if it is similar to a diagonal matrix. In other words, a diagonal matrix A has the property of an invertible matrix P and a diagonal matrix D such that A = PDP−1.
Discussion of Paper ‘Improved Explicit Integration Algorithms for Structural Dynamic Analysis with Unconditional Stability and Controllable Numerical Dissipation’ by Chinmoy Kolay & James M. Ricles, Journal of Earthquake Engineering 2017, http://www.tandfonline.com/loi/ueqe20
Published in Journal of Earthquake Engineering, 2021
Shuenn-Yih Chang, S. Veerarajan, Tsui-Huang Wu
where and can be found from for the specified initial conditions of and . In general, if the amplification matrix in the limiting case of has three linearly independent eigenvectors, then it is diagonalizable. As a result, one can have , where is an eigenvector matrix and is a diagonal matrix and its diagonal term , . As a result, the eigenvalues of the amplification matrix for the MKR-α method in the limit is found to be: