On eigenvectors and adjoints of modified matrices
L. Elsner, Péter Rózsa · Linear and Multilinear Algebra · 1981
We study how the adjoint (or adjugate) adj(A) of a real or complex n × n-matrix A behaves under the rank one modification A →A+uvH . The most interesting cases are rank A = n − 1, rank(A + uvH ) = n or n − 1. In the first case relations with generalized inverses can be shown, e.g. that (adj(A+uvH )−adj(A))/vH adj(A)u is a {1, 2}-inverse and that conversely any {1, 2}-inverse of A can be expressed in this form. The results in the second case can be interpreted as results on the changes of an eigenvector under this modification and can be carried over to matrices A without rank restrictions. Finally we investigate the adjoint of a bordered matrix.