Generalized eigenvectors

Linear Algebra Problems

Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018

When the generalized eigenvector matrix V is found, the Jordan canonical form can be obtained from J = V−1AV. It should be noted that the main diagonal elements in the Jordan matrix are the eigenvalues, with the main sub-diagonal elements taking 1’s.

Matrix Decomposition with Applications

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Published in Crista Arangala, Exploring Linear Algebra, 2019

Crista Arangala

If a matrix A is not diagonalizable it may be because the algebraic multiplicity of one or more eigenvalues is greater than 1 and the eigenvectors are linearly dependent. We can however find the Jordan canonical form of A using generalized eigenvectors.

Lamb waves in cubic crystals: crossing fundamental branches

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Journal Information

Published in Mechanics of Advanced Materials and Structures, 2023

Sergey V. Kuznetsov

However, if matrix is not semisimple, then (i) matrix contains both eigenvectors and generalized eigenvector(s); and, (ii) matrix along with diagonal part contains (one) Jordan block. The number of generalized eigenvectors depends upon the order of the Jordan block: one generalized eigenvector for the second order, and two generalized eigenvectors for the Jordan block of the third order [37].

Generalized eigenvectors

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Linear Algebra Problems

Matrix Decomposition with Applications

Lamb waves in cubic crystals: crossing fundamental branches