A determinantal lower bound

Bahman Kalantari, Thomas H. Pate

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

There are many known upper bounds for |det(·)|, the modulus of the determinant function, but useful lower bounds are rare. We show that if ℓ and u are positive lower and upper bounds on the moduli of the eigenvalues of a real or complex invertible n×n matrix A such that |tr(A)|≥nℓ, then |det(A)|≥ℓκun-κ, where κ=κ(ℓ,u)=[(nu-|tr(A)|)/(u-ℓ)]. Since κ≤n, our lower bound improves the obvious lower bound ℓn. In two cases the trace inequality is automatically satisfied. If A is a diagonally dominant real matrix with positive diagonal entries, then one can easily compute ℓ and u using Gerschgorin's circle theorem. While if A is positive definite Hermitian with eigenvalues λ1≤λ2≤≤λn, then our bound implies the inequality det(A)≥λ1κλnn-κ, where κ=κ(λ1n). Lower bounds like those established here have proved useful in obtaining new formulas for the approximation of π, e, square roots of numbers, and more generally real or complex roots of arbitrary polynomials.

Original languageEnglish (US)
Pages (from-to)151-159
Number of pages9
JournalLinear Algebra and Its Applications
Volume326
Issue number1-3
DOIs
StatePublished - Mar 15 2001

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Lower bound
Modulus
Circle theorem
Polynomials
Trace Inequality
Eigenvalue
Invertible matrix
Square root
Positive definite
Upper and Lower Bounds
Determinant
Roots
Upper bound
Imply
Polynomial
Arbitrary
Approximation

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis

Cite this

Kalantari, Bahman ; Pate, Thomas H. / A determinantal lower bound. In: Linear Algebra and Its Applications. 2001 ; Vol. 326, No. 1-3. pp. 151-159.
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A determinantal lower bound. / Kalantari, Bahman; Pate, Thomas H.

In: Linear Algebra and Its Applications, Vol. 326, No. 1-3, 15.03.2001, p. 151-159.

Research output: Contribution to journalArticle

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