Generalizations of the Ostrowski–Brauer theorem Original Research Article

The main theorem of this paper, which generalizes the Ostrowski–Brauer theorem and its previous extensions, provides conditions necessary and sufficient for the singularity of an irreducible matrix image satisfying the conditionsimageaiiajjgreater-or-equal, slantedRi(A)Rj(A),whereimageRk(A)=∑j≠kakj, k=1,…,n,for all i≠j such that aij+aji≠0 and implies a new description of the location of matrix eigenvalues in terms of ovals of Cassini and Gerschgorin circles.