Circular Hessenberg pairs

A square matrix is called Hessenberg whenever each entry below the subdiagonal zero and on nonzero. Let M denote a matrix. Then circular upper-right corner of nonzero every other above superdiagonal zero. pair consists two diagonalizable linear maps finite-dimensional vector space, that act an eigenbasis one in fashion. A,A⁎ pair. We investigate six bases for underlying space we find attractive. display transition matrices between certain pairs among six. also represent A⁎ with respect to bases. introduce special type pair, said be recurrent. show recurrent if only satisfy tridiagonal relations. For there related object system. classify up isomorphism systems. To this end, construct four families system isomorphic member families.