Abstract
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.
Highlights
There are many problems that make use of the eigenvalue or eigenvector of a matrix in their solution
Eigenvalue and eigenvectors are used throughout finite-element analysis (FEA) solutions to vibration problems, where the goal is often to minimize a structure’s sensitivity to various parameters through the use of eigenvalue/eigenvector derivatives [1]
A new formulation is derived for the complete Jacobians of eigenvalues and eigenvectors with respect to the elements of their parent matrix
Summary
There are many problems that make use of the eigenvalue or eigenvector of a matrix in their solution. Much of the literature on this topic is only valid when the parent matrix and its eigenvalues and eigenvectors are real due to the choice of the normalization of the eigenvectors Due to these observed deficiencies, the authors of this paper proposed a new method in [39] that did not involve the left eigenvector, did not require the simultaneous solution of both the eigenvector and eigenvalue derivatives, provided the full Jacobian matrices with respect to every element of the parent matrix, and provided a solution capable of handling any matrix with real or complex elements and real or complex eigenvalues and eigenvectors. The execution time on a digital computer is compared for the technique proposed in [39] and for the new technique proposed this paper
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