On weighted geometrically block diagonally cross dominant matrices (Q1568257)

The author considers a square complex block matrix \(A\) with square diagonal entries that are invertible. Such a matrix \(A\) gives rise to a block comparison matrix \(M(A)\) which is defined in terms of matrix norms of the submatrices and their inverses. Then \(A\) is said to be block (strictly) diagonally dominant, if the block comparison matrix \(M(A)\) is (strictly) diagonally dominant. Also, \(A\) is called quasi-block diagonally dominant (QBD) if there is a positive diagonal matrix \(Z\) such that \(AZ\) is block strictly diagonally dominant. Several conditions are given for \(A\) to be QBD. See \textit{M. Pang, G. Mao} [J. Math. Res. Expo. 11, No. 4, 507-513 (1991; Zbl 0787.15005] and \textit{T. Huang} [Math. Numer. Sin. 16, No. 1, 19-24 (1994; Zbl 0925.15018)] for related results.