Observations on the Computation of Eigenvalue and Eigenvector Jacobians
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 ar...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-11-01
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| Series: | Algorithms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1999-4893/12/12/245 |
| _version_ | 1811310576305963008 |
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| author | Andrew J. Liounis John A. Christian Shane B. Robinson |
| author_facet | Andrew J. Liounis John A. Christian Shane B. Robinson |
| author_sort | Andrew J. Liounis |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T10:01:15Z |
| format | Article |
| id | doaj.art-cb62745ac5fd4ababf7aea1be30fef12 |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-04-13T10:01:15Z |
| publishDate | 2019-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-cb62745ac5fd4ababf7aea1be30fef122022-12-22T02:51:14ZengMDPI AGAlgorithms1999-48932019-11-01121224510.3390/a12120245a12120245Observations on the Computation of Eigenvalue and Eigenvector JacobiansAndrew J. Liounis0John A. Christian1Shane B. Robinson2NASA Goddard Space Flight Center, Greenbelt, MD 20771, USARensselaer Polytechnic Institute, Troy, NY 12180, USANASA Johnson Space Center, Houston, TX 77058, USAMany 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.https://www.mdpi.com/1999-4893/12/12/245eigenvectoreigenvaluejacobian |
| spellingShingle | Andrew J. Liounis John A. Christian Shane B. Robinson Observations on the Computation of Eigenvalue and Eigenvector Jacobians Algorithms eigenvector eigenvalue jacobian |
| title | Observations on the Computation of Eigenvalue and Eigenvector Jacobians |
| title_full | Observations on the Computation of Eigenvalue and Eigenvector Jacobians |
| title_fullStr | Observations on the Computation of Eigenvalue and Eigenvector Jacobians |
| title_full_unstemmed | Observations on the Computation of Eigenvalue and Eigenvector Jacobians |
| title_short | Observations on the Computation of Eigenvalue and Eigenvector Jacobians |
| title_sort | observations on the computation of eigenvalue and eigenvector jacobians |
| topic | eigenvector eigenvalue jacobian |
| url | https://www.mdpi.com/1999-4893/12/12/245 |
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