Matrices whose inverses are generalizedM-matrices
Reinhard Nabben · Linear and Multilinear Algebra · 1998
In 1972 T. L. Markham defined the so-called matrices of type-D [M]. Markham proved that the inverse of a certain type-D matrix is a tridiagonal M-matrix. Recently, several generalizations of this result were established, i.e., other classes of inverse M-matrices were introduced. Here we give another generalization. We consider block matrices whose blocks are hermitian positive semidefinite which leads to block type-D matrices and block generalized ultrametric matrices. We will establish that the inverse of a block generalized ultrametric matrix is a generalized M-matrix. Moreover, we give an explicit formula for the inverse of a block type-D matrix which shows that the inverse is a block tridiagonal generalized M-matrix.