Diagonalizable matrix – Knowledge and References

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Introduction

Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018

If A is a diagonalizable matrix, namely, A∼D, where D is a diagonal matrix of eigenvalues, then A=SDS-1. A function of the matrix A is defined as f(A)=Sf(D)S-1.To implement the diagonalization procedure, do the following steps:

Topics from Linear Algebra

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Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022

Vladimir A. Dobrushkin

If A is a diagonalizable matrix, namely, A~Λ, where Λ is a diagonal matrix of eigenvalues, then A=SΛS−1. A function of the matrix A is defined as f(A)=Sf(Λ)S−1.

Eigenvalues and eigenvectors

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Published in Qingwen Hu, Concise Introduction to Linear Algebra, 2017

Qingwen Hu

12. Let A be an n × n diagonalizable matrix. Show that if every eigenvalue of A satisfies |λ|<1 $ |\lambda | < 1 $ , then

General formulation of macro-elements for the simulation of multi-layered bonded structures

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Journal Information

Published in The Journal of Adhesion, 2020

Sébastien Schwartz, Eric Paroissien, Frédéric Lachaud

The aforementioned method wants to explain the analytical solution of a matrix system of differential equations. Taking into consideration the system Equation (15), the matrix A is a squared matrix n x n, with n = 12. It exists an invertible matrix P such as J = P−1.A.P where J is the Jordan normal form.[35–40] According to[41–43], it corresponds to a generalization of the diagonalization procedure where a diagonalizable matrix is a particular case. Thus, any squared matrix has a Jordan normal form.[44] Therefore, Equation (15) becomes: