A state-of-the-art review on theory and engineering applications of eigenvalue and eigenvector derivatives

2020 ◽  
Vol 138 ◽  
pp. 106536
Author(s):  
R.M. Lin ◽  
J.E. Mottershead ◽  
T.Y. Ng
Keyword(s):  
State Of The Art ◽  
Engineering Applications ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector

scholarly journals Observations on the Computation of Eigenvalue and Eigenvector Jacobians

Algorithms ◽  
2019 ◽  
Vol 12 (12) ◽  
pp. 245 ◽  
Author(s):  
Andrew J. Liounis ◽  
John A. Christian ◽  
Shane B. Robinson
Keyword(s):  
Complex Case ◽  
Extensive Literature ◽  
Jacobian Matrices ◽  
Special Cases ◽  
The Matrix ◽  
General Complex ◽  
Analytic Expressions ◽  
And Performance ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector

Many scientific and engineering problems benefit from analytic expressions for eigenvalue and eigenvector derivatives with respect to the elements of the parent matrix. While there exists extensive literature on the calculation of these derivatives, which take the form of Jacobian matrices, there are a variety of deficiencies that have yet to be addressed—including the need for both left and right eigenvectors, limitations on the matrix structure, and issues with complex eigenvalues and eigenvectors. This work addresses these deficiencies by proposing a new analytic solution for the eigenvalue and eigenvector derivatives. The resulting analytic Jacobian matrices are numerically efficient to compute and are valid for the general complex case. It is further shown that this new general result collapses to previously known relations for the special cases of real symmetric matrices and real diagonal matrices. Finally, the new Jacobian expressions are validated using forward finite differencing and performance is compared with another technique.

Eigenvalue and eigenvector derivatives of a nondefective matrix

1989 ◽  
Vol 12 (4) ◽  
pp. 480-486 ◽  
Author(s):  
Jer-Nan Juang ◽  
Peiman Ghaemmaghami ◽  
Kyong Been Lim
Keyword(s):  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector ◽  
Derivatives Of

Eigenvector Derivatives of Generalized Nondefective Eigenproblems With Repeated Eigenvalues

1995 ◽  
Vol 117 (1) ◽  
pp. 207-212 ◽  
Author(s):  
Y.-Q. Zhang ◽  
W.-L. Wang
Keyword(s):  
Recent Work ◽  
New Method ◽  
First Eigenvalue ◽  
Numerical Examples ◽  
Repeated Eigenvalues ◽  
Generalized Eigenproblem ◽  
Eigenvalue Derivatives ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector ◽  
Derivatives Of

A new method is presented for computation of eigenvalue and eigenvector derivatives associated with repeated eigenvalues of the generalized nondefective eigenproblem. This approach is an extension of recent work by Daily and by Juang et al. and is applicable to symmetric or nonsymmetric systems. The extended phases read as follows. The differentiable eigenvectors and their derivatives associated with repeated eigenvalues are determined for a generalized eigenproblem, requiring the knowledge of only those eigenvectors to be differentiated. Moreover, formulations for computing eigenvector derivatives have been presented covering the case where multigroups of repeated first eigenvalue derivatives occur. Numerical examples are given to demonstrate the effectiveness of the proposed method.

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