# Observations on the Computation of Eigenvalue and Eigenvector Jacobians +
## 1 Introduction  
There are many problems that make use of the eigenvalue or eigenvector of a matrix in their solution. Because of this, it is often beneficial to be able to calculate the Jacobians of eigenvalues and eigenvectors with respect to the elements of the matrix from which they were computed. For example, 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]. As a second example, there are many instances in which the solution for the optimal estimate of a parameter vector takes the form of an eigenvalue problem--such as quaternion-based spacecraft attitude estimation [2; 3] or ellipse fitting in computer vision [4]. For these types of problems, it is often important to understand the uncertainty in the optimal estimate, which requires knowledge of the eigenvector Jacobian. Other examples abound, such as direction-of-arrival of plane waves [5], epipolar geometry in computer vision [6], pose estimation in robotics [7], spacecraft optical navigation [8], or infectious disease epidemiology [9]. In all of these problems, and in many others, it can be useful (or, in some cases, necessary) to have an analytic expression for the Jacobians of eigenvalues and eigenvectors with respect to the entries in their parent matrix. Because of the popularity of problems requiring the calculation of eigenvalues, eigenvectors, and their Jacobians, much work has already been devoted to these topics. The prior work on eigenvalue and eigenvector Jacobians can largely be grouped into three basic categories: modal expansion techniques, direct techniques, and techniques based on [PERSON]'s method. Modal techniques mimic the idea ofmodal expansion used in FEA, whereby the eigenvector Jacobian is expressed as a linear combination of the other eigenvectors of the system [10; 11; 12; 13; 14; 15; 16; 17]. Direct solutions find the Jacobians through the solution of a linear system of equations [18; 19; 20; 21; 22; 23; 24; 25]. Solutions based on [PERSON]'s method find the Jacobians by considering a homogeneous and particular solution [26; 27; 28; 29; 30; 31; 32; 33]. Finally, there is a grouping of solutions that are not easily fit into the above categories [34; 35; 36; 37; 38; 39]. For a more thorough discussion of this literature, see [39]. In light of the extensive literature already available on the topic, it may not be clear why more work is needed. After careful review, however, a few limitations are apparent with the existing methods. First, the majority of the above literature only presents solutions for the derivatives of the eigenvalues and eigenvectors with respect to a single element of the parent matrix. While these single element derivatives may be assembled to create a Jacobian, their use is cumbersome and the expressions are not compact. In the literature reviewed by these authors, expressions for the full Jacobians of the eigenvalues and eigenvectors only appear once and only for the special case of a symmetric parent matrix [34]. Second, in the majority of the literature, the derivatives are found using both the left and right eigenvectors of the system or by simultaneously solving a system of equations for both derivatives at once. While it is straightforward to find both the left and right eigensystems, calculating the left eigenvector in order to find the Jacobian of the right eigenvector is an unnecessary extra step that we seek to circumvent. Finally, 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.