Orthogonal square root eigenfactor parameterization of mass matrices
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abstract
An improved method is presented to parameterize a smoothly time varying, symmetric, positive definite system mass matrix M(t) in terms of the instantaneous eigenfactors, namely, the eigenvalues and eigenvectors of M(t). Differential equations are desired whose solutions generate the instantaneous spectral decomposition of M(t). The derivation makes use of the fact that the eigenvector matrix is orthogonal and, thus, evolves analogously to a higher-dimensional rotation matrix. Careful attention is given to cases where some eigenvalues and/or their derivatives are equal or near equal. A robust method is presented to approximate the corresponding eigenvector derivatives in these cases, which ensures that the resulting eigenvectors still diagnnalize the instantaneous M(t) matrix. This method is also capable of handling the rare case of discontinuous eigenvectors, which may only occur if both the corresponding eigenvalues and their derivatives are equal.
