Direct approach to the band completion problem
The paper deals with the partial band matrix completion problem, that is, given a partial band matrix \(A\), is there a completion \(A_c\) of \(A\) whose inverse is a band matrix? The problem considered in this paper concerns the completion of a partial block matrix with a specified band. The completion is required to be invertible and the inverse to be a band matrix of the same width as the specified band of the original matrix. The completion given by the authors is made directly via explicit formulas. They show that the solution of the band completion problem coincides with the problem of completion of a band to a Green matrix. This result is based on a theorem on relations between the ranks of submatrices of a block matrix and the ranks of the corresponding submatrices of its inverse. In addition, the completion obtained belongs to the class of quasiseparable matrices. At the end of the paper the authors study two special examples. The first is concerned with the Toeplitz case and the second contains a block tridiagonal case and the problem of completion of a special matrix which is connected with the problem of completions of triangular parts of matrices to contractions.
scientific article; zbMATH DE number 2083514
scientific article