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Zbl 1147.15018
Berenhaut, Kenneth S.; Guy, Richard T.; Vish, Nathaniel G.
A 1-norm bound for inverses of triangular matrices with monotone entries.
(English)
[J] Banach J. Math. Anal. 2, No. 1, 112-121, electronic only (2008). ISSN 1735-8787/e

Let $A= [a_{ij}]^n_{i,j=1}$ be an $n$-by-$n$ lower-triangular real matrix with $a_{ii}\ge a_{i+1,i}\ge\cdots\ge a_{n,i}> a> 0$ for all $i$, and $a_{11}\le a_{22}\le\cdots\le a_{nn}$. The main result of this paper gives an upper bound for the 1-norm of the inverse matrix of $A$, namely, $$\Vert A^{-1}\Vert_1\le{1\over a}\,\Biggl({a_{nn}\over a_{11}}\Biggr) (2-\rho(a, a_{nn})^{\lceil n/2\rceil}- \rho(a, a_{nn})^{\lfloor n/2\rfloor}),$$ where $\rho$ is defined as $\rho(x, y)= 1- (x/y)$. For the case of $a_{11}= a_{22}=\cdots= a_{nn}$, 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.
[Pei Yuan Wu (Hsinchu)]
MSC 2000:
*15A60 Appl. of functional analysis to matrix theory
15A09 Matrix inversion
15A45 Miscellaneous inequalities involving matrices
15A57 Other types of matrices

Keywords: inverse matrix; monotone entries; triangular matrix; partial order; recurrence relation; upper bound for the 1-norm; Toeplitz matrices

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