On tridiagonal conjugate secondary normal matrices

Con-s-normal matrices play the same role in the theory of s-unitary congruences as conventional s-normal matrices do with respect to s-unitary similarities. Naturally, the properties of both matrix classes are fairly similar up to the distinction between the congruence and similarity. However, in certain respects, con-s-normal matrices differ substantially from s-normal ones. Our goal in this paper is to indicate one of such distinctions. It is shown that none of the familiar characterizations of s-normal matrices having the irreducible tridiagonal form has a natural counterpart in the case of con-s-normal matrices. AMS Classification: 15A21, 15A09, 15457.