ON DIFFERENTIATING E!GENVALUES AND EIG ENVECTORS
Jan R. Magnus · 1985
Let XO be a square matrix (complex or otherwise) and uo a (normalized) eigenvector associated with an eigenvalue ,0 of XO, so that the triple (Xo, uo, L0) satisfies the equations Xu = )lu, u*u = 1. We investigate the conditions under which unique differentiable functions A(X) and u(X) exist in a neighborhood of XO satisfying 4(XO) = 40, u(Xo) = uo, Xu = Au, and u*u = 1. We obtain the first and second derivatives of A(X) and the first derivative of u(X). Two alternative expressions for the first derivative of A(X) are also presented.