Normal matrices and their principal submatrices of co-order one

Let A be a normal matrix, v be any of its indices, A-v be the matrix obtained from A by deleting the vth row and column, and λ be an eigenvalue of A-v. In our paper we construct the eigenspace of A associated with λ from that of A-v. In particular, it is shown that if there is a (unique) Jordan block of size strictly greater than one in the part of the Jordan form of A-v corresponding to λ, then the geometric multiplicity of λ decreases by one under the transition from A-v to A (in other words, the typical change of the spectral properties holds for λ). The results obtained are applied to circulant matrices. Moreover, in Appendix to our paper we consider almost regular tournament matrices as principal submatrices of co-order one of regular tournament matrices. In particular, it is observed that the Brualdi-Li tournament matrix B2n of order 2n is permutationally similar to a principal submatrix of co-order one of the circulant matrix of order 2n + 1 with the first row . As a consequence of this fact, the weak Brualdi-Li conjecture is formulated for principal submatrices of co-order one of the adjacency matrices of Cayley tournaments.