On covariances of eigenvalues and eigenvectors of second-rank symmetric tensors

Abstract

S U M M A R YThe 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).

Why Does Theoretical Physics Fail to Explain and Predict Earthquake Occurrence?

“…Although tensors are fundamentally important, they have not been su ciently investigated or interpreted from a statistical point of view in the earth sciences, with few exceptions. A linear error propagation was rst independently proposed to derive the error estimate of the principal stresses and their orientations by Angelier et al (1982) and Soler and van Gelder (1991). The correlation study on the invariant quantities of seismic moment tensors was investigated by Kagan and Knopo (1985a b).…”

Statistical analysis of geodetic deformation (strain rate) derived from the space geodetic measurements of BIFROST Project in Fennoscandia

“…The first work on the statistical analysis of random tensors in the Earth Sciences was to compute the first-order accuracy of the principal eigenvalues of a symmetric, rank-two random deformation tensor (Angelier et al, 1982 as an appendix, and probably independently, Soler and van Gelder, 1991;Feigl et al, 1990). Kagan and Knopoff (1985) studied statistically the first two moments of stochastic three-dimensional (3D) seismic moment tensor invariants, which were used to explain complex fault geometry (Kagan, 1992).…”

On covariance propagation of eigenparameters of symmetric n-D tensors

“…Angelier et al 1982 and Wyss et al 1992) utilized this approach, although, they restricted their investigation to determine the estimates of the eigenvalues. In addition, Soler & van Gelder (1991) extended the formulation to compute the covariance matrices of the eigenvalues and their principal directions. This theory was discovered not to be fully general, and an extension was incorporated into an erratum published in Soler & van Gelder (2006).…”