Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]