Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem

2007 ◽  
Vol 16 ◽  
Author(s):  
N. Van der Aa ◽  
H. Ter Morsche ◽  
R. Mattheij
Keyword(s):  
General Complex ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector ◽  
Complex Valued

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.

scholarly journals A Numerical Method for Computing the Roots of Non-Singular Complex-Valued Matrices

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 966
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Numerical Method ◽  
Worked Examples ◽  
Higher Order ◽  
Singular Matrix ◽  
The Matrix ◽  
General Complex ◽  
Complex Valued

A method for the computation of the n th roots of a general complex-valued r × r non-singular matrix ? is presented. The proposed procedure is based on the Dunford–Taylor integral (also ascribed to Riesz–Fantappiè) and relies, only, on the knowledge of the invariants of the matrix, so circumventing the computation of the relevant eigenvalues. Several worked examples are illustrated to validate the developed algorithm in the case of higher order matrices.

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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