On sufficient conditions for the existence of a unitary congruence transformation of a given complex matrix into a real one

A complex n × n matrix A is said to be nonderogatory if the degree of its minimal polynomial is equal to the degree of the characteristic polynomial. The aim of the paper is to prove the following assertion: Let A`(A)] A\bar{A} be a nonderogatory matrix with real positive spectrum. Then A can be made real by a unitary congruence transformation if and only if A and `(A)] \bar{A} are unitarily congruent. Bibliography: 5 titles.