On the inverse of band matrices

Band matrices and Green's matrices have been defined by E. Asplund [1]; his basic theorem shows their relationship. The definition of one-pair-matrices was given by F. R. Gantmacher and M. G. Krein [5]; they showed that the inverse of a one-pair-matrix is a tridiagonal matrix and vice versa. Introducing the concept of strict band matrices and semi-separable matrices, and making use of the above theorems, the main result on the relationship between strict band matrices and semi-separable matrices will be proved. Further a constructive method will be given to obtain the inverse of a strict band matrix.