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

Let 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.