### 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)|≥ℓ^{κ}u^{n-κ}, where κ=κ(ℓ,u)=[(nu-|tr(A)|)/(u-ℓ)]. Since κ≤n, our lower bound improves the obvious lower bound ℓ^{n}. 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 λ_{1}≤λ_{2}≤≤λ_{n}, then our bound implies the inequality det(A)≥λ_{1}^{κ}λ_{n} ^{n-κ}, where κ=κ(λ_{1},λ_{n}). 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 language | English (US) |
---|---|

Pages (from-to) | 151-159 |

Number of pages | 9 |

Journal | Linear Algebra and Its Applications |

Volume | 326 |

Issue number | 1-3 |

DOIs | |

State | Published - Mar 15 2001 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Keywords

- Determinants
- Eigenvalues
- Optimization

### Cite this

*Linear Algebra and Its Applications*,

*326*(1-3), 151-159. https://doi.org/10.1016/S0024-3795(00)00287-1

}

*Linear Algebra and Its Applications*, vol. 326, no. 1-3, pp. 151-159. https://doi.org/10.1016/S0024-3795(00)00287-1

**A determinantal lower bound.** / Kalantari, Bahman; Pate, Thomas H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A determinantal lower bound

AU - Kalantari, Bahman

AU - Pate, Thomas H.

PY - 2001/3/15

Y1 - 2001/3/15

N2 - 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)≥λ1κλn n-κ, where κ=κ(λ1,λn). 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.

AB - 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)≥λ1κλn n-κ, where κ=κ(λ1,λn). 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.

KW - Determinants

KW - Eigenvalues

KW - Optimization

UR - http://www.scopus.com/inward/record.url?scp=0035582262&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035582262&partnerID=8YFLogxK

U2 - 10.1016/S0024-3795(00)00287-1

DO - 10.1016/S0024-3795(00)00287-1

M3 - Article

AN - SCOPUS:0035582262

VL - 326

SP - 151

EP - 159

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -