On Multiple Eigenvalues of Matrices Depending on Several Parameters
King-wah Eric Chu · SIAM Journal on Numerical Analysis · 1990
The eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by Sun [J. Comput. Math., 3 (1985), pp. 351–364] (using the implicit function theorems) and others, but not for the more general multiple eigenvalues. In this paper, the mean of the multiple eigenvalues (in terms of the trace operator tr), the corresponding invariant subspaces, and their derivatives are studied, using a generalization of Sun’s approach. The numerical aspects concerning the computation of such derivatives by direct and iterative methods (e.g., simultaneous and inverse simultaneous iterations) will be discussed briefly. The main results in this paper apply to clusters of nonmultiple eigenvalues as well. The implications, e.g., on simultaneous iteration, will be explored.