On inverses of infinite Hessenberg matrices

Here a known result on the structure of finite Hessenberg matrices is extended to infinite Hessenberg matrices. Its consequences for the example of infinite Hessenberg–Toeplitz matrices are described. The results are applied also to the inversion of infinite tridiagonal matrices via recurrence relations. Moreover, since there are available free parameters, different inverses can be associated with a given invertible tridiagonal matrix.