On the change of the Jordan form under the transition from the adjacency matrix of a vertex-transitive digraph to its principal submatrix of co-order one

AbstractLet J(λ;n1,…,nk) be the set of matrices A such that λ is an eigenvalue of A and n1⩽⋯⩽nk are the sizes of the Jordan blocks associated with λ. For a given index v of A, denote by A−v the principal submatrix of co-order one obtained from A by deleting the vth row and column. In the present paper, all possible changes of the part of the Jordan form corresponding to λ under the transition from A to A−v are determined for matrices A∈J(λ;n1,…,nk) such that for the eigenvalue λ of both A and A⊤, there exists a Jordan chain of the largest length nk whose eigenvector has nonzero vth entry. In particular, it is shown that for almost every matrix A∈J(λ;n1,…,nk), n1,…,nk−1 are the sizes of Jordan blocks for λ considered as an eigenvalue of A−v. Moreover, it is also proved that if A is the adjacency matrix of a vertex-transitive digraph and k⩾2, then the change n1,…,nk→n1,…,nk−2,2nk−1−1 holds for the eigenvalue λ under the transition from A to A−v. In the case of k=1, λ is a simple eigenvalue of A and does not belong to the spectrum of A−v