Let A be a normal matrix, v be any of its indices, A - v be the matrix obtained from A by deleting the v th 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 B 2 n of order 2 n is permutationally similar to a principal submatrix of co-order one of the circulant matrix of order 2 n + 1 with the first row 0 , 1 , … , 1 ︸ n , 0 , … , 0 ︸ n . 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.