Block LU factorization is stable for block matrices whose inverses are block diagonally dominant

Abstract

Let A ∈ M_n (C) and let the inverse matrix B = A⁻¹ be block diagonally dominant by rows (columns) w.r.t. an m × m block partitioning and a matrix norm. We show that A possesses a block LU factorization w.r.t. the same block partitioning, and the growth factor for A in this factorization is bounded above by 1 + σ, where σ = max _1≤i≤m σ_i and σ_i, 0 ≤ σ_i ≤ 1, are the row (column) block dominance factors of B. Further, the off-diagonal blocks of A (and of its block Schur complements) satisfy the inequalities

$$\parallel A_{ji} A_{ii}^{ - 1} \parallel \;\; \leqslant \sigma _j ,\;\;\;\;j \ne i.$$

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