Geophys. J. fnf. (1991) 105, 537-546 RESEARCH NOTE On covariances of eigenvalues and eigenvectors of second-rank symmetric tensors Tomas Soler’ and Boudewijn H. W. van Gelder2 National Geodetic Survey, Charting and Geodetic Services, National Ocean Service, NOAA , Rockville, MD 20852, USA Faculty of Geodetic Engineering, Dew University of Technology, Thijsseweg 11, 2629 JA Delft, The Netherlaah Accepted 1990 November 22. Received 1990 November 22; in original form 1990 June 14 SUMMARY The applications of eigentheory to many branches of mathematical physics (e.g., rotational dynamics, continuum mechanics) is an unquestionable fact. This work expands the conventional methodology by introducing equations to compute the covariance matrices of eigenvalues and eigenvectors of second-rank 3-D symmetric tensors in terms of their six distinct elements error estimates. New analytical expressions derived herein are general and should be of interest to anyone concerned with the accuracy of the computed orientation of principal axes and their associated principal quantities (e.g., moments of inertia, stress, strain). Key words: eigenvalues, eigenvectors, principal strain, principal stress. INTRODUCTION In mechanics, dynamics, statistics, etc. symmetric 3-D second-rank tensors play important roles. They are used to describe the mechanical properties of rigid or deformable bodies in terms of inertia, stress or strain. In addition, the statistical properties of the 3-D position of points on those bodies represented by covariance matrices, or its geometric interpretation in terms of ‘error ellipsoids’ are often required. The equations to transform, under a known rotation, the components of these types of tensors between two arbitrary Cartesian coordinate frames are well known. Among all possible coordinate frames, emphasis is generally placed upon the one defining the principal axes. Through the use of eigenvalues and eigenvectors 3 x 3 symmetric tensors consisting of six distinct elements can be diagonalized by choosing an appropriate reference frame. The original tensor will transform into a tensor of diagonal form (three eigenvalues), representing e.g., principal quantities (moments of inertia, strain, stress), and a rotation (eigenvector) matrix, specifying the rotation of the original arbitrary reference frame into the coordinate frame in which the mechanical or statistical properties become ‘principal’ or ‘uncorrelated’. In this note formulae to assess the precision of the components of the final rotated tensor as a function of the precision estimates of the components of the original given tensor are first established. Then, novel analytical expressions to compute the covariance matrix ,of the eigenvalues and eigenvectors in terms of the same initial variables are developed. This in turn provides the accuracies of important physical parameters such as magnitude and orientation of principal moments of inertia, strain, stress, etc. The literature currently available discusses thoroughly the computation and properties of eigenvalues and eigenvectors of 3-D symmetric tensors but systematically ignores their accuracies. This paper supplements many textbooks in linear algebra and statistics by answering the question of how reliable the computed eigenvalues and eigenvectors are as implied from the known estimates of accuracy associated with the original undiagonalized tensor components. COVARIANCE MATRIX OF ROTATED SYMMETRIC TENSOR COMPONENTS Restricting the present discussion to the 3-D space of our ordinary experience, assume that a transformation of the type [&’I = R[&]R‘ (1) 531 by guest on January 26, 2013 http://gji.oxfordjournals.org/ Downloaded from