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A 1-norm bound for inverses of triangular matrices with monotone entries. (English) Zbl 1147.15018

Let A=[a ij ] i,j=1 n be an n-by-n lower-triangular real matrix with a ii a i+1,i a n,i >a>0 for all i, and a 11 a 22 a nn . The main result of this paper gives an upper bound for the 1-norm of the inverse matrix of A, namely,

A -1 1 1 aa nn a 11 (2-ρ(a,a nn ) n/2 -ρ(a,a nn ) n/2 ),

where ρ is defined as ρ(x,y)=1-(x/y). For the case of a 11 =a 22 ==a nn , the inequality is shown to be best possible. The result extends and refines the previous ones for (lower-triangular) Toeplitz matrices. The proof is elementary but slightly lengthy.

MSC:
15A60Applications of functional analysis to matrix theory
15A09Matrix inversion, generalized inverses
15A45Miscellaneous inequalities involving matrices
15A57Other types of matrices (MSC2000)