Some notes on Z-matrices
This paper deals with new results on Z- and M-matrices. First, a irreducibility criterion is proved: For a matrix being partitioned into four blocks with one being nonsingular in the main diagonal and those being in the second diagonal being nonzero, if the Schur complement of the matrix being nonsingular is irreducible, then the overall matrix is irreducible. Furthermore, if the overall matrix is an irreducible Z-matrix and the nonsingular block matrix is an M-matrix then the Schur complement of such a block matrix is also irreducible. An inequality is obtained relating the eigenvalue of minimum modulus of an \(N_ 0\)-matrix to the eigenvalue of minimum modulus of the Schur complement in that matrix of an M- submatrix. Similar results are obtained for irreducible M-matrices and irreducible nonsingular \(F_ 0\)-matrices. Circulant \(N_ 0\)-matrices of order three are characterized in terms of their spectrum while matrices of \(F_ 0\) inverses are shown to have nonpositive principal minors of order \(\geq 2\).
scientific article; zbMATH DE number 4063282