Some remarks on a theorem of Gudkov

Diagonal dominance, Schur complements and some classes of H-matrices and P-matrices

“…The previous nonsingularity condition was given by Gudkov in [8], although Ostrowski credited Nekrasov the discovery of this condition, and so matrices satisfying this condition are known in the literature as Nekrasov matrices. As an immediate consequence of Corollary 2 of [18], we have that a Nekrasov matrix is an H-matrix.…”

“…This class of SDD 1 matrices generalizes strict diagonal dominance by rows in the sense that rows very strictly diagonally dominant can guarantee that the matrix is nonsingular despite other rows are not diagonally dominant. This idea had been already used in the concept of Nekrasov matrices (see [7,8,18]), but, in contrast to the class of matrices considered in this paper, Nekrasov matrices required that the first row is strictly diagonally dominant. In this sense, SDD 1 by rows only requires that the matrix has at least a strictly diagonally dominant row, as in general H-matrices, but not necessarily the first one.…”

“…Matrices SDD 1 by rows share with the class of matrices SDD by rows and the H-matrices the invariance under symmetric permutations (i.e., under simultaneous coincident row and column exchanges or, equivalently, A belongs to the class if and only if PAP T belongs to it, for any permutation matrix P), in contrast to other intermediate classes of matrices between SDD by rows and H-matrices, such as Nekrasov matrices (see [6][7][8]18]). …”

Scaling technique for Partition-Nekrasov matrices

“…The original setting was in terms of the convergence of the Gauss-Seidel iteration (see [15]). This class of matrices was further discussed in many papers and it was used to obtain max-norm bounds of the inverse, bounds for determinants, and, also, this class was a starting point for many different generalisations, made in order to expand this nonsingularity result to wider classes of matrices (see [5][6][7]16,19,24]). Because of the way Nekrasov class is defined, involving recursively calculated row sums, finding the whole class of corresponding diagonal scaling matrices (as it is done for Partition-SDD) is not an easy task.…”

Loewner matrix ordering in estimation of the smallest singular value

“…Before stating Gudkov's nonsingularity condition [1] we mention that matrices that satisfy it are known in the literature as Nekrasov matrices ( [12]). …”