Open AccessOn covariance propagation of eigenparameters of symmetric n‐D tensors
Author(s) -
Han J. Y.,
Van Gelder B. H. W.,
Soler T.
Publication year - 2007
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.2007.03416.x
Subject(s) - eigenvalues and eigenvectors , tensor (intrinsic definition) , covariance , symmetric tensor , mathematics , principal component analysis , invariants of tensors , covariance matrix , cartesian tensor , tensor calculus , linear algebra , cauchy stress tensor , tensor density , mathematical analysis , pure mathematics , physics , geometry , tensor field , algorithm , statistics , exact solutions in general relativity , quantum mechanics
SUMMARY Symmetric tensors are typically encountered during investigations associated with stress and strain analysis and, thus, they are of particular interest to geophysicists and geodesists. Furthermore, symmetric tensors are studied using eigentheory analysis which provides the decomposition of the tensor on its principal components (n independent eigenvalues and the corresponding eigenvectors). In this paper, an analytical expression of the covariance matrix of the eigenvalues and eigenvectors of an n‐D symmetric tensor is derived based on the principles of linear algebra and differential calculus. Through numerical tests, the proposed formulation is proven to give realistic uncertainty estimates of the determined eigenparameters. The methodology also reveals the significant impact on uncertainty assessments when the parameter dependencies between principal components are neglected.